Average Error: 33.0 → 10.8
Time: 22.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r737066 = b_2;
        double r737067 = -r737066;
        double r737068 = r737066 * r737066;
        double r737069 = a;
        double r737070 = c;
        double r737071 = r737069 * r737070;
        double r737072 = r737068 - r737071;
        double r737073 = sqrt(r737072);
        double r737074 = r737067 + r737073;
        double r737075 = r737074 / r737069;
        return r737075;
}
↓
double f(double a, double b_2, double c) {
        double r737076 = b_2;
        double r737077 = -9.348931433494438e+39;
        bool r737078 = r737076 <= r737077;
        double r737079 = a;
        double r737080 = r737076 / r737079;
        double r737081 = -2.0;
        double r737082 = 0.5;
        double r737083 = c;
        double r737084 = r737082 * r737083;
        double r737085 = r737084 / r737076;
        double r737086 = fma(r737080, r737081, r737085);
        double r737087 = 1.3353078790738604e-121;
        bool r737088 = r737076 <= r737087;
        double r737089 = r737076 * r737076;
        double r737090 = r737079 * r737083;
        double r737091 = r737089 - r737090;
        double r737092 = sqrt(r737091);
        double r737093 = r737092 - r737076;
        double r737094 = r737093 / r737079;
        double r737095 = 1.6168702840263923e-79;
        bool r737096 = r737076 <= r737095;
        double r737097 = -0.5;
        double r737098 = r737083 / r737076;
        double r737099 = r737097 * r737098;
        double r737100 = 1.546013236023957e-67;
        bool r737101 = r737076 <= r737100;
        double r737102 = r737101 ? r737094 : r737099;
        double r737103 = r737096 ? r737099 : r737102;
        double r737104 = r737088 ? r737094 : r737103;
        double r737105 = r737078 ? r737086 : r737104;
        return r737105;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    3. Taylor expanded around -inf 6.2

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

    4. Simplified6.2

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

    if -9.348931433494438e+39 < b_2 < 1.3353078790738604e-121 or 1.6168702840263923e-79 < b_2 < 1.546013236023957e-67

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-inv13.0

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied un-div-inv12.8

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

    if 1.3353078790738604e-121 < b_2 < 1.6168702840263923e-79 or 1.546013236023957e-67 < b_2

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 11.2

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2} \cdot c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))