Average Error: 19.3 → 6.3
Time: 17.4s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.119950078133889 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\]
\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 -2.0519886074223697 \cdot 10^{+155}:\\
\;\;\;\;\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}\\

\mathbf{elif}\;b \le 8.119950078133889 \cdot 10^{+99}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \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{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r907051 = b;
        double r907052 = 0.0;
        bool r907053 = r907051 >= r907052;
        double r907054 = 2.0;
        double r907055 = c;
        double r907056 = r907054 * r907055;
        double r907057 = -r907051;
        double r907058 = r907051 * r907051;
        double r907059 = 4.0;
        double r907060 = a;
        double r907061 = r907059 * r907060;
        double r907062 = r907061 * r907055;
        double r907063 = r907058 - r907062;
        double r907064 = sqrt(r907063);
        double r907065 = r907057 - r907064;
        double r907066 = r907056 / r907065;
        double r907067 = r907057 + r907064;
        double r907068 = r907054 * r907060;
        double r907069 = r907067 / r907068;
        double r907070 = r907053 ? r907066 : r907069;
        return r907070;
}
↓
double f(double a, double b, double c) {
        double r907071 = b;
        double r907072 = -2.0519886074223697e+155;
        bool r907073 = r907071 <= r907072;
        double r907074 = 0.0;
        bool r907075 = r907071 >= r907074;
        double r907076 = 2.0;
        double r907077 = c;
        double r907078 = r907076 * r907077;
        double r907079 = -r907071;
        double r907080 = r907071 * r907071;
        double r907081 = 4.0;
        double r907082 = a;
        double r907083 = r907081 * r907082;
        double r907084 = r907083 * r907077;
        double r907085 = r907080 - r907084;
        double r907086 = sqrt(r907085);
        double r907087 = r907079 - r907086;
        double r907088 = r907078 / r907087;
        double r907089 = r907071 / r907077;
        double r907090 = r907082 / r907089;
        double r907091 = r907090 - r907071;
        double r907092 = r907076 * r907091;
        double r907093 = r907076 * r907082;
        double r907094 = r907092 / r907093;
        double r907095 = r907075 ? r907088 : r907094;
        double r907096 = 8.119950078133889e+99;
        bool r907097 = r907071 <= r907096;
        double r907098 = sqrt(r907086);
        double r907099 = r907098 * r907098;
        double r907100 = r907079 - r907099;
        double r907101 = r907078 / r907100;
        double r907102 = r907086 + r907079;
        double r907103 = r907102 / r907093;
        double r907104 = r907075 ? r907101 : r907103;
        double r907105 = r907078 / r907092;
        double r907106 = r907082 * r907077;
        double r907107 = r907106 / r907071;
        double r907108 = r907107 - r907071;
        double r907109 = r907076 * r907108;
        double r907110 = r907109 / r907093;
        double r907111 = r907075 ? r907105 : r907110;
        double r907112 = r907097 ? r907104 : r907111;
        double r907113 = r907073 ? r907095 : r907112;
        return r907113;
}

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 < -2.0519886074223697e+155

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 11.5

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

    3. Simplified2.7

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

    if -2.0519886074223697e+155 < b < 8.119950078133889e+99

    1. Initial program 8.4

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

    3. Applied add-sqr-sqrt8.4

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

    4. Applied sqrt-prod8.5

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

    if 8.119950078133889e+99 < b

    1. Initial program 29.4

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

    2. Taylor expanded around inf 6.8

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

    3. Simplified2.3

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

    4. Taylor expanded around -inf 2.3

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

    5. Simplified2.3

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.0519886074223697 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.119950078133889 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))))