Average Error: 33.8 → 8.7
Time: 17.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 3.849558639383284492876426675330012960339 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.357284433132494781653830699845499522001 \cdot 10^{61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -a, 0\right)}{\sqrt{{b_2}^{2} - a \cdot c} + b_2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\

\mathbf{elif}\;b_2 \le 3.849558639383284492876426675330012960339 \cdot 10^{-181}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}{a} - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.357284433132494781653830699845499522001 \cdot 10^{61}:\\
\;\;\;\;\frac{\mathsf{fma}\left(c, -a, 0\right)}{\sqrt{{b_2}^{2} - a \cdot c} + b_2} \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r23112 = b_2;
        double r23113 = -r23112;
        double r23114 = r23112 * r23112;
        double r23115 = a;
        double r23116 = c;
        double r23117 = r23115 * r23116;
        double r23118 = r23114 - r23117;
        double r23119 = sqrt(r23118);
        double r23120 = r23113 + r23119;
        double r23121 = r23120 / r23115;
        return r23121;
}
↓
double f(double a, double b_2, double c) {
        double r23122 = b_2;
        double r23123 = -8.301687926884189e+98;
        bool r23124 = r23122 <= r23123;
        double r23125 = c;
        double r23126 = r23125 / r23122;
        double r23127 = 0.5;
        double r23128 = -2.0;
        double r23129 = a;
        double r23130 = r23122 / r23129;
        double r23131 = r23128 * r23130;
        double r23132 = fma(r23126, r23127, r23131);
        double r23133 = 3.8495586393832845e-181;
        bool r23134 = r23122 <= r23133;
        double r23135 = r23129 * r23125;
        double r23136 = -r23135;
        double r23137 = fma(r23122, r23122, r23136);
        double r23138 = sqrt(r23137);
        double r23139 = r23138 / r23129;
        double r23140 = r23139 - r23130;
        double r23141 = 1.3572844331324948e+61;
        bool r23142 = r23122 <= r23141;
        double r23143 = -r23129;
        double r23144 = 0.0;
        double r23145 = fma(r23125, r23143, r23144);
        double r23146 = 2.0;
        double r23147 = pow(r23122, r23146);
        double r23148 = r23147 - r23135;
        double r23149 = sqrt(r23148);
        double r23150 = r23149 + r23122;
        double r23151 = r23145 / r23150;
        double r23152 = 1.0;
        double r23153 = r23152 / r23129;
        double r23154 = r23151 * r23153;
        double r23155 = -0.5;
        double r23156 = r23155 * r23126;
        double r23157 = r23142 ? r23154 : r23156;
        double r23158 = r23134 ? r23140 : r23157;
        double r23159 = r23124 ? r23132 : r23158;
        return r23159;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -8.301687926884189e+98

    1. Initial program 46.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified46.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied fma-neg46.2

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}} - b_2}{a}\]

    5. Taylor expanded around -inf 3.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    6. Simplified3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)}\]

    if -8.301687926884189e+98 < b_2 < 3.8495586393832845e-181

    1. Initial program 10.8

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified10.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied fma-neg10.8

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}} - b_2}{a}\]
    5. Using strategy rm

    6. Applied div-sub10.8

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}{a} - \frac{b_2}{a}}\]

    if 3.8495586393832845e-181 < b_2 < 1.3572844331324948e+61

    1. Initial program 34.8

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified34.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied fma-neg34.8

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}} - b_2}{a}\]
    5. Using strategy rm

    6. Applied flip--34.9

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)} - b_2 \cdot b_2}{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)} + b_2}}}{a}\]

    7. Simplified15.6

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c, -a, 0\right)}}{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)} + b_2}}{a}\]

    8. Simplified15.6

      \[\leadsto \frac{\frac{\mathsf{fma}\left(c, -a, 0\right)}{\color{blue}{\sqrt{{b_2}^{2} - a \cdot c} + b_2}}}{a}\]
    9. Using strategy rm

    10. Applied div-inv15.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, -a, 0\right)}{\sqrt{{b_2}^{2} - a \cdot c} + b_2} \cdot \frac{1}{a}}\]

    if 1.3572844331324948e+61 < b_2

    1. Initial program 57.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified57.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 4.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 3.849558639383284492876426675330012960339 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.357284433132494781653830699845499522001 \cdot 10^{61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -a, 0\right)}{\sqrt{{b_2}^{2} - a \cdot c} + b_2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))