Average Error: 33.2 → 9.9
Time: 9.2m
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.2295616480632551 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.303853124735619 \cdot 10^{+50}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.2295616480632551 \cdot 10^{-79}:\\
\;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r18126000 = b_2;
        double r18126001 = -r18126000;
        double r18126002 = r18126000 * r18126000;
        double r18126003 = a;
        double r18126004 = c;
        double r18126005 = r18126003 * r18126004;
        double r18126006 = r18126002 - r18126005;
        double r18126007 = sqrt(r18126006);
        double r18126008 = r18126001 + r18126007;
        double r18126009 = r18126008 / r18126003;
        return r18126009;
}
↓
double f(double a, double b_2, double c) {
        double r18126010 = b_2;
        double r18126011 = -3.303853124735619e+50;
        bool r18126012 = r18126010 <= r18126011;
        double r18126013 = 0.5;
        double r18126014 = c;
        double r18126015 = r18126014 / r18126010;
        double r18126016 = r18126013 * r18126015;
        double r18126017 = a;
        double r18126018 = r18126010 / r18126017;
        double r18126019 = r18126016 - r18126018;
        double r18126020 = r18126019 - r18126018;
        double r18126021 = 1.2295616480632551e-79;
        bool r18126022 = r18126010 <= r18126021;
        double r18126023 = 1.0;
        double r18126024 = r18126023 / r18126017;
        double r18126025 = r18126010 * r18126010;
        double r18126026 = r18126017 * r18126014;
        double r18126027 = r18126025 - r18126026;
        double r18126028 = sqrt(r18126027);
        double r18126029 = r18126024 * r18126028;
        double r18126030 = r18126029 - r18126018;
        double r18126031 = -0.5;
        double r18126032 = r18126015 * r18126031;
        double r18126033 = r18126022 ? r18126030 : r18126032;
        double r18126034 = r18126012 ? r18126020 : r18126033;
        return r18126034;
}

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 < -3.303853124735619e+50

    1. Initial program 35.2

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

    2. Simplified35.2

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

    4. Applied div-sub35.2

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

    5. Taylor expanded around -inf 5.1

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

    if -3.303853124735619e+50 < b_2 < 1.2295616480632551e-79

    1. Initial program 12.6

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

    2. Simplified12.6

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

    4. Applied div-sub12.6

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

    6. Applied *-un-lft-identity12.6

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

    7. Applied associate-/l*12.6

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}} - \frac{b_2}{a}\]
    8. Using strategy rm

    9. Applied div-inv12.7

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

    10. Applied *-un-lft-identity12.7

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

    11. Applied times-frac12.7

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

    12. Simplified12.7

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

    if 1.2295616480632551e-79 < b_2

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.2295616480632551 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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))