Average Error: 34.2 → 6.5
Time: 7.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.6038168240882645 \cdot 10^{144}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.2731438419880699 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.1125387673008883 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{2}{4}}}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.6038168240882645 \cdot 10^{144}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.2731438419880699 \cdot 10^{-203}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.1125387673008883 \cdot 10^{122}:\\
\;\;\;\;\frac{\frac{\frac{1}{\frac{2}{4}}}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r55147 = b;
        double r55148 = -r55147;
        double r55149 = r55147 * r55147;
        double r55150 = 4.0;
        double r55151 = a;
        double r55152 = c;
        double r55153 = r55151 * r55152;
        double r55154 = r55150 * r55153;
        double r55155 = r55149 - r55154;
        double r55156 = sqrt(r55155);
        double r55157 = r55148 + r55156;
        double r55158 = 2.0;
        double r55159 = r55158 * r55151;
        double r55160 = r55157 / r55159;
        return r55160;
}
↓
double f(double a, double b, double c) {
        double r55161 = b;
        double r55162 = -7.603816824088264e+144;
        bool r55163 = r55161 <= r55162;
        double r55164 = 1.0;
        double r55165 = c;
        double r55166 = r55165 / r55161;
        double r55167 = a;
        double r55168 = r55161 / r55167;
        double r55169 = r55166 - r55168;
        double r55170 = r55164 * r55169;
        double r55171 = -3.27314384198807e-203;
        bool r55172 = r55161 <= r55171;
        double r55173 = -r55161;
        double r55174 = r55161 * r55161;
        double r55175 = 4.0;
        double r55176 = r55167 * r55165;
        double r55177 = r55175 * r55176;
        double r55178 = r55174 - r55177;
        double r55179 = sqrt(r55178);
        double r55180 = sqrt(r55179);
        double r55181 = r55180 * r55180;
        double r55182 = r55173 + r55181;
        double r55183 = 2.0;
        double r55184 = r55183 * r55167;
        double r55185 = r55182 / r55184;
        double r55186 = 2.1125387673008883e+122;
        bool r55187 = r55161 <= r55186;
        double r55188 = 1.0;
        double r55189 = r55183 / r55175;
        double r55190 = r55188 / r55189;
        double r55191 = r55188 / r55165;
        double r55192 = r55190 / r55191;
        double r55193 = r55173 - r55179;
        double r55194 = r55192 / r55193;
        double r55195 = -1.0;
        double r55196 = r55195 * r55166;
        double r55197 = r55187 ? r55194 : r55196;
        double r55198 = r55172 ? r55185 : r55197;
        double r55199 = r55163 ? r55170 : r55198;
        return r55199;
}

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

Target

Original34.2
Target21.2
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.603816824088264e+144

    1. Initial program 61.2

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -7.603816824088264e+144 < b < -3.27314384198807e-203

    1. Initial program 7.1

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

    3. Applied add-sqr-sqrt7.1

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

    4. Applied sqrt-prod7.4

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

    if -3.27314384198807e-203 < b < 2.1125387673008883e+122

    1. Initial program 29.8

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

    3. Applied flip-+29.9

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

    4. Simplified16.2

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity16.2

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

    7. Applied *-un-lft-identity16.2

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]

    8. Applied times-frac16.2

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    9. Applied associate-/l*16.3

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]

    10. Simplified15.5

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]
    11. Using strategy rm

    12. Applied associate-/r*15.3

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

    13. Simplified9.5

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

    if 2.1125387673008883e+122 < b

    1. Initial program 61.0

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

    2. Taylor expanded around inf 2.1

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.6038168240882645 \cdot 10^{144}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.2731438419880699 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.1125387673008883 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{2}{4}}}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))