Average Error: 19.1 → 7.2
Time: 14.4s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6844644503075447 \cdot 10^{+144}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \le -1.1253904323982412 \cdot 10^{-267}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{\left(a \cdot 4\right) \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 3.060834910674178 \cdot 10^{+83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.6844644503075447 \cdot 10^{+144}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le -1.1253904323982412 \cdot 10^{-267}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\frac{\left(a \cdot 4\right) \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 3.060834910674178 \cdot 10^{+83}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r497060 = b;
        double r497061 = 0.0;
        bool r497062 = r497060 >= r497061;
        double r497063 = 2.0;
        double r497064 = c;
        double r497065 = r497063 * r497064;
        double r497066 = -r497060;
        double r497067 = r497060 * r497060;
        double r497068 = 4.0;
        double r497069 = a;
        double r497070 = r497068 * r497069;
        double r497071 = r497070 * r497064;
        double r497072 = r497067 - r497071;
        double r497073 = sqrt(r497072);
        double r497074 = r497066 - r497073;
        double r497075 = r497065 / r497074;
        double r497076 = r497066 + r497073;
        double r497077 = r497063 * r497069;
        double r497078 = r497076 / r497077;
        double r497079 = r497062 ? r497075 : r497078;
        return r497079;
}
↓
double f(double a, double b, double c) {
        double r497080 = b;
        double r497081 = -1.6844644503075447e+144;
        bool r497082 = r497080 <= r497081;
        double r497083 = 0.0;
        bool r497084 = r497080 >= r497083;
        double r497085 = 2.0;
        double r497086 = -r497080;
        double r497087 = r497080 * r497080;
        double r497088 = a;
        double r497089 = 4.0;
        double r497090 = r497088 * r497089;
        double r497091 = c;
        double r497092 = r497090 * r497091;
        double r497093 = r497087 - r497092;
        double r497094 = sqrt(r497093);
        double r497095 = r497086 - r497094;
        double r497096 = r497095 / r497091;
        double r497097 = r497085 / r497096;
        double r497098 = r497091 / r497080;
        double r497099 = r497080 / r497088;
        double r497100 = r497098 - r497099;
        double r497101 = r497084 ? r497097 : r497100;
        double r497102 = -1.1253904323982412e-267;
        bool r497103 = r497080 <= r497102;
        double r497104 = r497085 * r497091;
        double r497105 = r497094 - r497080;
        double r497106 = r497092 / r497105;
        double r497107 = r497104 / r497106;
        double r497108 = r497086 + r497094;
        double r497109 = r497088 * r497085;
        double r497110 = r497108 / r497109;
        double r497111 = r497084 ? r497107 : r497110;
        double r497112 = 3.060834910674178e+83;
        bool r497113 = r497080 <= r497112;
        double r497114 = 1.0;
        double r497115 = r497114 / r497095;
        double r497116 = r497104 * r497115;
        double r497117 = r497084 ? r497116 : r497100;
        double r497118 = r497080 / r497091;
        double r497119 = r497088 / r497118;
        double r497120 = r497119 - r497080;
        double r497121 = r497120 * r497085;
        double r497122 = r497104 / r497121;
        double r497123 = r497084 ? r497122 : r497100;
        double r497124 = r497113 ? r497117 : r497123;
        double r497125 = r497103 ? r497111 : r497124;
        double r497126 = r497082 ? r497101 : r497125;
        return r497126;
}

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 4 regimes

  2. if b < -1.6844644503075447e+144

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 10.0

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

    3. Simplified2.6

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

    4. Taylor expanded around 0 2.5

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

    6. Applied associate-/l*2.5

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

    if -1.6844644503075447e+144 < b < -1.1253904323982412e-267

    1. Initial program 7.8

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

    3. Applied add-sqr-sqrt7.8

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

    4. Applied sqrt-prod7.8

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

    6. Applied flip--7.8

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

    7. Simplified7.8

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

    8. Simplified7.8

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

    if -1.1253904323982412e-267 < b < 3.060834910674178e+83

    1. Initial program 8.7

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

    2. Taylor expanded around -inf 12.1

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

    3. Simplified12.1

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

    4. Taylor expanded around 0 12.1

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

    6. Applied div-inv12.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

    if 3.060834910674178e+83 < b

    1. Initial program 28.8

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

    2. Taylor expanded around -inf 28.8

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

    3. Simplified28.8

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

    4. Taylor expanded around 0 28.8

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

    5. Taylor expanded around inf 6.8

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

    6. Simplified2.6

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6844644503075447 \cdot 10^{+144}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \le -1.1253904323982412 \cdot 10^{-267}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{\left(a \cdot 4\right) \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 3.060834910674178 \cdot 10^{+83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\left(2 \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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))))