Average Error: 20.3 → 7.7
Time: 9.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le -2.394215615302178077989514942164219717454 \cdot 10^{-310} \lor \neg \left(b \le 2.976536699626750437221780308255562179441 \cdot 10^{116}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}\\

\mathbf{elif}\;b \le -2.394215615302178077989514942164219717454 \cdot 10^{-310} \lor \neg \left(b \le 2.976536699626750437221780308255562179441 \cdot 10^{116}\right):\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r31127 = b;
        double r31128 = 0.0;
        bool r31129 = r31127 >= r31128;
        double r31130 = 2.0;
        double r31131 = c;
        double r31132 = r31130 * r31131;
        double r31133 = -r31127;
        double r31134 = r31127 * r31127;
        double r31135 = 4.0;
        double r31136 = a;
        double r31137 = r31135 * r31136;
        double r31138 = r31137 * r31131;
        double r31139 = r31134 - r31138;
        double r31140 = sqrt(r31139);
        double r31141 = r31133 - r31140;
        double r31142 = r31132 / r31141;
        double r31143 = r31133 + r31140;
        double r31144 = r31130 * r31136;
        double r31145 = r31143 / r31144;
        double r31146 = r31129 ? r31142 : r31145;
        return r31146;
}
↓
double f(double a, double b, double c) {
        double r31147 = b;
        double r31148 = -2.6152573735422387e+153;
        bool r31149 = r31147 <= r31148;
        double r31150 = 0.0;
        bool r31151 = r31147 >= r31150;
        double r31152 = 2.0;
        double r31153 = c;
        double r31154 = r31152 * r31153;
        double r31155 = 1.0;
        double r31156 = -r31147;
        double r31157 = a;
        double r31158 = r31157 * r31153;
        double r31159 = r31158 / r31147;
        double r31160 = r31152 * r31159;
        double r31161 = r31147 - r31160;
        double r31162 = r31156 - r31161;
        double r31163 = r31155 / r31162;
        double r31164 = r31154 * r31163;
        double r31165 = 1.0;
        double r31166 = r31153 / r31147;
        double r31167 = r31147 / r31157;
        double r31168 = r31166 - r31167;
        double r31169 = r31165 * r31168;
        double r31170 = r31151 ? r31164 : r31169;
        double r31171 = -2.3942156153022e-310;
        bool r31172 = r31147 <= r31171;
        double r31173 = 2.9765366996267504e+116;
        bool r31174 = r31147 <= r31173;
        double r31175 = !r31174;
        bool r31176 = r31172 || r31175;
        double r31177 = 2.0;
        double r31178 = r31177 * r31147;
        double r31179 = r31160 - r31178;
        double r31180 = r31154 / r31179;
        double r31181 = r31147 * r31147;
        double r31182 = 4.0;
        double r31183 = r31182 * r31157;
        double r31184 = r31183 * r31153;
        double r31185 = r31181 - r31184;
        double r31186 = sqrt(r31185);
        double r31187 = r31156 + r31186;
        double r31188 = r31152 * r31157;
        double r31189 = r31187 / r31188;
        double r31190 = r31151 ? r31180 : r31189;
        double r31191 = r31156 - r31186;
        double r31192 = r31154 / r31191;
        double r31193 = r31182 * r31158;
        double r31194 = r31193 / r31191;
        double r31195 = r31194 / r31188;
        double r31196 = r31151 ? r31192 : r31195;
        double r31197 = r31176 ? r31190 : r31196;
        double r31198 = r31149 ? r31170 : r31197;
        return r31198;
}

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 < -2.6152573735422387e+153

    1. Initial program 63.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around -inf 10.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    3. Taylor expanded around 0 2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\\ \end{array}\]

    4. Simplified2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
    5. Using strategy rm

    6. Applied div-inv2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    7. Taylor expanded around inf 2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    if -2.6152573735422387e+153 < b < -2.3942156153022e-310 or 2.9765366996267504e+116 < b

    1. Initial program 18.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 8.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    if -2.3942156153022e-310 < b < 2.9765366996267504e+116

    1. Initial program 8.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied flip-+8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Simplified8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le -2.394215615302178077989514942164219717454 \cdot 10^{-310} \lor \neg \left(b \le 2.976536699626750437221780308255562179441 \cdot 10^{116}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))