Average Error: 34.4 → 10.0
Time: 5.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.91243958875386880555748684589545292526 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.209120745343099452134664059704875392955 \cdot 10^{-70}:\\ \;\;\;\;\frac{0 - 1 \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.91243958875386880555748684589545292526 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.209120745343099452134664059704875392955 \cdot 10^{-70}:\\
\;\;\;\;\frac{0 - 1 \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r14767 = b_2;
        double r14768 = -r14767;
        double r14769 = r14767 * r14767;
        double r14770 = a;
        double r14771 = c;
        double r14772 = r14770 * r14771;
        double r14773 = r14769 - r14772;
        double r14774 = sqrt(r14773);
        double r14775 = r14768 + r14774;
        double r14776 = r14775 / r14770;
        return r14776;
}
↓
double f(double a, double b_2, double c) {
        double r14777 = b_2;
        double r14778 = -9.912439588753869e+101;
        bool r14779 = r14777 <= r14778;
        double r14780 = 0.5;
        double r14781 = c;
        double r14782 = r14781 / r14777;
        double r14783 = r14780 * r14782;
        double r14784 = 2.0;
        double r14785 = a;
        double r14786 = r14777 / r14785;
        double r14787 = r14784 * r14786;
        double r14788 = r14783 - r14787;
        double r14789 = 1.2091207453430995e-70;
        bool r14790 = r14777 <= r14789;
        double r14791 = 0.0;
        double r14792 = 1.0;
        double r14793 = r14777 * r14777;
        double r14794 = r14785 * r14781;
        double r14795 = r14793 - r14794;
        double r14796 = sqrt(r14795);
        double r14797 = r14777 - r14796;
        double r14798 = r14792 * r14797;
        double r14799 = r14791 - r14798;
        double r14800 = r14799 / r14785;
        double r14801 = -0.5;
        double r14802 = r14801 * r14782;
        double r14803 = r14790 ? r14800 : r14802;
        double r14804 = r14779 ? r14788 : r14803;
        return r14804;
}

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 3 regimes

  2. if b_2 < -9.912439588753869e+101

    1. Initial program 46.9

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

    2. Taylor expanded around -inf 3.7

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

    if -9.912439588753869e+101 < b_2 < 1.2091207453430995e-70

    1. Initial program 13.3

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

    3. Applied neg-sub013.3

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

    4. Applied associate-+l-13.3

      \[\leadsto \frac{\color{blue}{0 - \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity13.3

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

    if 1.2091207453430995e-70 < b_2

    1. Initial program 53.7

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.91243958875386880555748684589545292526 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.209120745343099452134664059704875392955 \cdot 10^{-70}:\\ \;\;\;\;\frac{0 - 1 \cdot \left(b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))