Average Error: 19.2 → 6.7
Time: 18.0s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}} \cdot \left(\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}} \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.960333965284851 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{a \cdot 2}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 -1.7512236628315378 \cdot 10^{+131}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}} \cdot \left(\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}} \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}\\

\mathbf{elif}\;b \le 2.960333965284851 \cdot 10^{+101}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r487762 = b;
        double r487763 = 0.0;
        bool r487764 = r487762 >= r487763;
        double r487765 = 2.0;
        double r487766 = c;
        double r487767 = r487765 * r487766;
        double r487768 = -r487762;
        double r487769 = r487762 * r487762;
        double r487770 = 4.0;
        double r487771 = a;
        double r487772 = r487770 * r487771;
        double r487773 = r487772 * r487766;
        double r487774 = r487769 - r487773;
        double r487775 = sqrt(r487774);
        double r487776 = r487768 - r487775;
        double r487777 = r487767 / r487776;
        double r487778 = r487768 + r487775;
        double r487779 = r487765 * r487771;
        double r487780 = r487778 / r487779;
        double r487781 = r487764 ? r487777 : r487780;
        return r487781;
}
↓
double f(double a, double b, double c) {
        double r487782 = b;
        double r487783 = -1.7512236628315378e+131;
        bool r487784 = r487782 <= r487783;
        double r487785 = 0.0;
        bool r487786 = r487782 >= r487785;
        double r487787 = 2.0;
        double r487788 = c;
        double r487789 = r487787 * r487788;
        double r487790 = -r487782;
        double r487791 = a;
        double r487792 = r487791 * r487787;
        double r487793 = r487782 / r487788;
        double r487794 = r487792 / r487793;
        double r487795 = r487782 - r487794;
        double r487796 = r487790 - r487795;
        double r487797 = r487789 / r487796;
        double r487798 = cbrt(r487797);
        double r487799 = r487798 * r487798;
        double r487800 = r487798 * r487799;
        double r487801 = r487788 / r487782;
        double r487802 = r487782 / r487791;
        double r487803 = r487801 - r487802;
        double r487804 = r487786 ? r487800 : r487803;
        double r487805 = 2.960333965284851e+101;
        bool r487806 = r487782 <= r487805;
        double r487807 = r487782 * r487782;
        double r487808 = 4.0;
        double r487809 = r487791 * r487808;
        double r487810 = r487809 * r487788;
        double r487811 = r487807 - r487810;
        double r487812 = cbrt(r487811);
        double r487813 = fabs(r487812);
        double r487814 = sqrt(r487812);
        double r487815 = r487813 * r487814;
        double r487816 = r487790 - r487815;
        double r487817 = r487789 / r487816;
        double r487818 = sqrt(r487811);
        double r487819 = r487818 + r487790;
        double r487820 = r487819 / r487792;
        double r487821 = r487786 ? r487817 : r487820;
        double r487822 = sqrt(r487818);
        double r487823 = r487822 * r487822;
        double r487824 = r487790 + r487823;
        double r487825 = r487824 / r487792;
        double r487826 = r487786 ? r487797 : r487825;
        double r487827 = r487806 ? r487821 : r487826;
        double r487828 = r487784 ? r487804 : r487827;
        return r487828;
}

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 < -1.7512236628315378e+131

    1. Initial program 51.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 51.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified51.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    4. Taylor expanded around -inf 9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    5. Simplified2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \end{array}\]

    6. Taylor expanded around 0 2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{\left(\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}} \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}}\right) \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

    if -1.7512236628315378e+131 < b < 2.960333965284851e+101

    1. Initial program 8.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    5. Simplified9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right|} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    if 2.960333965284851e+101 < b

    1. Initial program 30.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 6.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    6. Applied sqrt-prod2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}} \cdot \left(\sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}} \cdot \sqrt[3]{\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.960333965284851 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))