Average Error: 34.2 → 9.1
Time: 5.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -21584226661136.2148:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.27945410118768703 \cdot 10^{-95}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.7411875700484855 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -21584226661136.2148:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.27945410118768703 \cdot 10^{-95}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\

\mathbf{elif}\;b_2 \le -2.125553485370055 \cdot 10^{-113}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 6.7411875700484855 \cdot 10^{112}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17664 = b_2;
        double r17665 = -r17664;
        double r17666 = r17664 * r17664;
        double r17667 = a;
        double r17668 = c;
        double r17669 = r17667 * r17668;
        double r17670 = r17666 - r17669;
        double r17671 = sqrt(r17670);
        double r17672 = r17665 - r17671;
        double r17673 = r17672 / r17667;
        return r17673;
}
↓
double f(double a, double b_2, double c) {
        double r17674 = b_2;
        double r17675 = -21584226661136.215;
        bool r17676 = r17674 <= r17675;
        double r17677 = -0.5;
        double r17678 = c;
        double r17679 = r17678 / r17674;
        double r17680 = r17677 * r17679;
        double r17681 = -3.279454101187687e-95;
        bool r17682 = r17674 <= r17681;
        double r17683 = 1.0;
        double r17684 = a;
        double r17685 = 0.0;
        double r17686 = r17684 * r17678;
        double r17687 = r17685 + r17686;
        double r17688 = r17674 * r17674;
        double r17689 = r17688 - r17686;
        double r17690 = sqrt(r17689);
        double r17691 = r17690 - r17674;
        double r17692 = r17687 / r17691;
        double r17693 = r17684 / r17692;
        double r17694 = r17683 / r17693;
        double r17695 = -2.125553485370055e-113;
        bool r17696 = r17674 <= r17695;
        double r17697 = 6.7411875700484855e+112;
        bool r17698 = r17674 <= r17697;
        double r17699 = -r17674;
        double r17700 = r17699 - r17690;
        double r17701 = r17684 / r17700;
        double r17702 = r17683 / r17701;
        double r17703 = 0.5;
        double r17704 = r17703 * r17679;
        double r17705 = 2.0;
        double r17706 = r17674 / r17684;
        double r17707 = r17705 * r17706;
        double r17708 = r17704 - r17707;
        double r17709 = r17698 ? r17702 : r17708;
        double r17710 = r17696 ? r17680 : r17709;
        double r17711 = r17682 ? r17694 : r17710;
        double r17712 = r17676 ? r17680 : r17711;
        return r17712;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -21584226661136.215 or -3.279454101187687e-95 < b_2 < -2.125553485370055e-113

    1. Initial program 54.8

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

    2. Taylor expanded around -inf 6.3

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

    if -21584226661136.215 < b_2 < -3.279454101187687e-95

    1. Initial program 38.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num38.2

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

    5. Applied add-exp-log40.0

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

    7. Applied flip--40.0

      \[\leadsto \frac{1}{\frac{a}{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \cdot e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{\left(-b_2\right) + e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}}}\]

    8. Simplified17.2

      \[\leadsto \frac{1}{\frac{a}{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}}\]

    9. Simplified14.9

      \[\leadsto \frac{1}{\frac{a}{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}\]

    if -2.125553485370055e-113 < b_2 < 6.7411875700484855e+112

    1. Initial program 12.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num12.3

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

    if 6.7411875700484855e+112 < b_2

    1. Initial program 49.8

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

    2. Taylor expanded around inf 2.8

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -21584226661136.2148:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.27945410118768703 \cdot 10^{-95}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.7411875700484855 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))