Average Error: 19.5 → 7.4
Time: 18.7s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.448476596858411 \cdot 10^{+162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.448476596858411 \cdot 10^{+162}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\\

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

\end{array}\\

\mathbf{elif}\;b \le 2.0410715251838527 \cdot 10^{+49}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r1329013 = b;
        double r1329014 = 0.0;
        bool r1329015 = r1329013 >= r1329014;
        double r1329016 = -r1329013;
        double r1329017 = r1329013 * r1329013;
        double r1329018 = 4.0;
        double r1329019 = a;
        double r1329020 = r1329018 * r1329019;
        double r1329021 = c;
        double r1329022 = r1329020 * r1329021;
        double r1329023 = r1329017 - r1329022;
        double r1329024 = sqrt(r1329023);
        double r1329025 = r1329016 - r1329024;
        double r1329026 = 2.0;
        double r1329027 = r1329026 * r1329019;
        double r1329028 = r1329025 / r1329027;
        double r1329029 = r1329026 * r1329021;
        double r1329030 = r1329016 + r1329024;
        double r1329031 = r1329029 / r1329030;
        double r1329032 = r1329015 ? r1329028 : r1329031;
        return r1329032;
}
↓
double f(double a, double b, double c) {
        double r1329033 = b;
        double r1329034 = -9.448476596858411e+162;
        bool r1329035 = r1329033 <= r1329034;
        double r1329036 = 0.0;
        bool r1329037 = r1329033 >= r1329036;
        double r1329038 = -r1329033;
        double r1329039 = r1329033 * r1329033;
        double r1329040 = 4.0;
        double r1329041 = a;
        double r1329042 = r1329040 * r1329041;
        double r1329043 = c;
        double r1329044 = r1329042 * r1329043;
        double r1329045 = r1329039 - r1329044;
        double r1329046 = sqrt(r1329045);
        double r1329047 = log(r1329046);
        double r1329048 = exp(r1329047);
        double r1329049 = r1329038 - r1329048;
        double r1329050 = 2.0;
        double r1329051 = r1329041 * r1329050;
        double r1329052 = r1329049 / r1329051;
        double r1329053 = r1329043 * r1329050;
        double r1329054 = r1329041 / r1329033;
        double r1329055 = r1329043 * r1329054;
        double r1329056 = r1329055 - r1329033;
        double r1329057 = r1329050 * r1329056;
        double r1329058 = r1329053 / r1329057;
        double r1329059 = r1329037 ? r1329052 : r1329058;
        double r1329060 = 2.0410715251838527e+49;
        bool r1329061 = r1329033 <= r1329060;
        double r1329062 = r1329038 - r1329046;
        double r1329063 = r1329062 / r1329051;
        double r1329064 = r1329041 * r1329043;
        double r1329065 = r1329040 * r1329064;
        double r1329066 = r1329039 - r1329065;
        double r1329067 = sqrt(r1329066);
        double r1329068 = sqrt(r1329067);
        double r1329069 = r1329068 * r1329068;
        double r1329070 = r1329069 + r1329038;
        double r1329071 = r1329053 / r1329070;
        double r1329072 = r1329037 ? r1329063 : r1329071;
        double r1329073 = -2.0;
        double r1329074 = r1329073 * r1329055;
        double r1329075 = r1329033 + r1329074;
        double r1329076 = r1329038 - r1329075;
        double r1329077 = r1329076 / r1329051;
        double r1329078 = r1329037 ? r1329077 : r1329058;
        double r1329079 = r1329061 ? r1329072 : r1329078;
        double r1329080 = r1329035 ? r1329059 : r1329079;
        return r1329080;
}

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 < -9.448476596858411e+162

    1. Initial program 39.6

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

    2. Taylor expanded around 0 39.6

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

    3. Simplified39.6

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

    5. Applied add-sqr-sqrt39.6

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

    6. Applied sqrt-prod39.6

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

    7. Taylor expanded around -inf 7.1

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

    8. Simplified1.1

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

    10. Applied add-exp-log1.1

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

    if -9.448476596858411e+162 < b < 2.0410715251838527e+49

    1. Initial program 9.2

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

    2. Taylor expanded around 0 9.1

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

    3. Simplified9.1

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

    5. Applied add-sqr-sqrt9.1

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

    6. Applied sqrt-prod9.3

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

    if 2.0410715251838527e+49 < b

    1. Initial program 36.3

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

    2. Taylor expanded around 0 36.3

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

    3. Simplified36.3

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

    5. Applied add-sqr-sqrt36.3

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

    6. Applied sqrt-prod36.3

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

    7. Taylor expanded around -inf 36.3

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

    8. Simplified36.3

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

    9. Taylor expanded around inf 11.1

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

    10. Simplified6.4

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.448476596858411 \cdot 10^{+162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - e^{\log \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))