Average Error: 19.1 → 12.9
Time: 35.3s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.7222948455593476 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}\]
\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 1.7222948455593476 \cdot 10^{+64}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r740136 = b;
        double r740137 = 0.0;
        bool r740138 = r740136 >= r740137;
        double r740139 = 2.0;
        double r740140 = c;
        double r740141 = r740139 * r740140;
        double r740142 = -r740136;
        double r740143 = r740136 * r740136;
        double r740144 = 4.0;
        double r740145 = a;
        double r740146 = r740144 * r740145;
        double r740147 = r740146 * r740140;
        double r740148 = r740143 - r740147;
        double r740149 = sqrt(r740148);
        double r740150 = r740142 - r740149;
        double r740151 = r740141 / r740150;
        double r740152 = r740142 + r740149;
        double r740153 = r740139 * r740145;
        double r740154 = r740152 / r740153;
        double r740155 = r740138 ? r740151 : r740154;
        return r740155;
}
↓
double f(double a, double b, double c) {
        double r740156 = b;
        double r740157 = 1.7222948455593476e+64;
        bool r740158 = r740156 <= r740157;
        double r740159 = 0.0;
        bool r740160 = r740156 >= r740159;
        double r740161 = 2.0;
        double r740162 = c;
        double r740163 = r740161 * r740162;
        double r740164 = -r740156;
        double r740165 = -4.0;
        double r740166 = a;
        double r740167 = r740166 * r740162;
        double r740168 = r740156 * r740156;
        double r740169 = fma(r740165, r740167, r740168);
        double r740170 = sqrt(r740169);
        double r740171 = r740164 - r740170;
        double r740172 = r740163 / r740171;
        double r740173 = sqrt(r740170);
        double r740174 = r740173 * r740173;
        double r740175 = r740164 + r740174;
        double r740176 = r740161 * r740166;
        double r740177 = r740175 / r740176;
        double r740178 = r740160 ? r740172 : r740177;
        double r740179 = r740162 / r740156;
        double r740180 = r740166 * r740179;
        double r740181 = r740180 - r740156;
        double r740182 = r740161 * r740181;
        double r740183 = r740163 / r740182;
        double r740184 = r740170 - r740156;
        double r740185 = r740184 / r740176;
        double r740186 = r740160 ? r740183 : r740185;
        double r740187 = r740158 ? r740178 : r740186;
        return r740187;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1.7222948455593476e+64

    1. Initial program 16.3

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

    2. Simplified16.3

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

    4. Applied add-sqr-sqrt16.3

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

    5. Applied sqrt-prod16.4

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

    7. Applied sub-neg16.4

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

    if 1.7222948455593476e+64 < b

    1. Initial program 26.3

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

    2. Simplified26.3

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

    4. Applied add-sqr-sqrt26.4

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

    5. Taylor expanded around inf 7.2

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

    6. Simplified3.7

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.7222948455593476 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(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))))