Average Error: 34.2 → 10.4
Time: 13.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \left(\frac{1}{a} \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.12310353364421125 \cdot 10^{95}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \left(\frac{1}{a} \cdot \frac{1}{2}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r42121 = b;
        double r42122 = -r42121;
        double r42123 = r42121 * r42121;
        double r42124 = 4.0;
        double r42125 = a;
        double r42126 = r42124 * r42125;
        double r42127 = c;
        double r42128 = r42126 * r42127;
        double r42129 = r42123 - r42128;
        double r42130 = sqrt(r42129);
        double r42131 = r42122 + r42130;
        double r42132 = 2.0;
        double r42133 = r42132 * r42125;
        double r42134 = r42131 / r42133;
        return r42134;
}
↓
double f(double a, double b, double c) {
        double r42135 = b;
        double r42136 = -4.123103533644211e+95;
        bool r42137 = r42135 <= r42136;
        double r42138 = 1.0;
        double r42139 = c;
        double r42140 = r42139 / r42135;
        double r42141 = a;
        double r42142 = r42135 / r42141;
        double r42143 = r42140 - r42142;
        double r42144 = r42138 * r42143;
        double r42145 = 3.446447862996811e-75;
        bool r42146 = r42135 <= r42145;
        double r42147 = r42135 * r42135;
        double r42148 = 4.0;
        double r42149 = r42148 * r42141;
        double r42150 = r42149 * r42139;
        double r42151 = r42147 - r42150;
        double r42152 = sqrt(r42151);
        double r42153 = r42152 - r42135;
        double r42154 = 1.0;
        double r42155 = r42154 / r42141;
        double r42156 = 2.0;
        double r42157 = r42154 / r42156;
        double r42158 = r42155 * r42157;
        double r42159 = r42153 * r42158;
        double r42160 = -1.0;
        double r42161 = r42160 * r42140;
        double r42162 = r42146 ? r42159 : r42161;
        double r42163 = r42137 ? r42144 : r42162;
        return r42163;
}

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 < -4.123103533644211e+95

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around -inf 3.8

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

    4. Simplified3.8

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

    if -4.123103533644211e+95 < b < 3.446447862996811e-75

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied *-un-lft-identity13.3

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

    5. Applied *-un-lft-identity13.3

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

    6. Applied times-frac13.3

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

    7. Applied associate-/l*13.4

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

    9. Applied div-inv13.4

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

    10. Applied *-un-lft-identity13.4

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

    11. Applied times-frac13.5

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

    12. Applied add-cube-cbrt13.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}\right) \cdot \sqrt[3]{\frac{1}{1}}}}{\frac{1}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \frac{a}{\frac{1}{2}}}\]

    13. Applied times-frac13.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}}{\frac{1}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\frac{a}{\frac{1}{2}}}}\]

    14. Simplified13.4

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\frac{a}{\frac{1}{2}}}\]

    15. Simplified13.4

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

    if 3.446447862996811e-75 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 9.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \left(\frac{1}{a} \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))