Average Error: 20.2 → 8.1
Time: 21.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.990473150623290141018988832760520387873 \cdot 10^{112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-2, b, \frac{2 \cdot \left(a \cdot c\right)}{b}\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 3.932191065039699403664081156929555460562 \cdot 10^{107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{2 \cdot a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\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 -1.990473150623290141018988832760520387873 \cdot 10^{112}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}\\

\mathbf{elif}\;b \le 3.932191065039699403664081156929555460562 \cdot 10^{107}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{2 \cdot a}{\frac{b}{c}}\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1010071 = b;
        double r1010072 = 0.0;
        bool r1010073 = r1010071 >= r1010072;
        double r1010074 = -r1010071;
        double r1010075 = r1010071 * r1010071;
        double r1010076 = 4.0;
        double r1010077 = a;
        double r1010078 = r1010076 * r1010077;
        double r1010079 = c;
        double r1010080 = r1010078 * r1010079;
        double r1010081 = r1010075 - r1010080;
        double r1010082 = sqrt(r1010081);
        double r1010083 = r1010074 - r1010082;
        double r1010084 = 2.0;
        double r1010085 = r1010084 * r1010077;
        double r1010086 = r1010083 / r1010085;
        double r1010087 = r1010084 * r1010079;
        double r1010088 = r1010074 + r1010082;
        double r1010089 = r1010087 / r1010088;
        double r1010090 = r1010073 ? r1010086 : r1010089;
        return r1010090;
}
↓
double f(double a, double b, double c) {
        double r1010091 = b;
        double r1010092 = -1.99047315062329e+112;
        bool r1010093 = r1010091 <= r1010092;
        double r1010094 = 0.0;
        bool r1010095 = r1010091 >= r1010094;
        double r1010096 = -r1010091;
        double r1010097 = r1010091 * r1010091;
        double r1010098 = 4.0;
        double r1010099 = a;
        double r1010100 = r1010098 * r1010099;
        double r1010101 = c;
        double r1010102 = r1010100 * r1010101;
        double r1010103 = r1010097 - r1010102;
        double r1010104 = sqrt(r1010103);
        double r1010105 = r1010096 - r1010104;
        double r1010106 = 2.0;
        double r1010107 = r1010106 * r1010099;
        double r1010108 = r1010105 / r1010107;
        double r1010109 = r1010101 * r1010106;
        double r1010110 = -2.0;
        double r1010111 = r1010099 * r1010101;
        double r1010112 = r1010106 * r1010111;
        double r1010113 = r1010112 / r1010091;
        double r1010114 = fma(r1010110, r1010091, r1010113);
        double r1010115 = r1010109 / r1010114;
        double r1010116 = r1010095 ? r1010108 : r1010115;
        double r1010117 = 3.9321910650396994e+107;
        bool r1010118 = r1010091 <= r1010117;
        double r1010119 = sqrt(r1010104);
        double r1010120 = cbrt(r1010104);
        double r1010121 = r1010120 * r1010120;
        double r1010122 = r1010121 * r1010120;
        double r1010123 = sqrt(r1010122);
        double r1010124 = r1010119 * r1010123;
        double r1010125 = r1010124 + r1010096;
        double r1010126 = r1010109 / r1010125;
        double r1010127 = r1010095 ? r1010108 : r1010126;
        double r1010128 = r1010091 / r1010101;
        double r1010129 = r1010107 / r1010128;
        double r1010130 = fma(r1010091, r1010110, r1010129);
        double r1010131 = r1010130 / r1010107;
        double r1010132 = r1010104 + r1010096;
        double r1010133 = r1010109 / r1010132;
        double r1010134 = r1010095 ? r1010131 : r1010133;
        double r1010135 = r1010118 ? r1010127 : r1010134;
        double r1010136 = r1010093 ? r1010116 : r1010135;
        return r1010136;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.99047315062329e+112

    1. Initial program 31.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt31.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    4. Applied sqrt-prod31.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    5. Taylor expanded around -inf 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

    6. Simplified6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(-2, b, \frac{2 \cdot \left(a \cdot c\right)}{b}\right)}}\\ \end{array}\]

    if -1.99047315062329e+112 < b < 3.9321910650396994e+107

    1. Initial program 9.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt9.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    4. Applied sqrt-prod9.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt9.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    if 3.9321910650396994e+107 < b

    1. Initial program 49.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt49.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Applied sqrt-prod49.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    5. Taylor expanded around inf 10.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    6. Simplified3.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{a \cdot 2}{\frac{b}{c}}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.990473150623290141018988832760520387873 \cdot 10^{112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-2, b, \frac{2 \cdot \left(a \cdot c\right)}{b}\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 3.932191065039699403664081156929555460562 \cdot 10^{107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{2 \cdot a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))