Average Error: 34.0 → 6.8
Time: 20.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.727723973148742563324153150500151191249 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.364529283357116165462040672443911208369 \cdot 10^{-246}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.198650993474429809884798868723112488797 \cdot 10^{74}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.727723973148742563324153150500151191249 \cdot 10^{105}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.364529283357116165462040672443911208369 \cdot 10^{-246}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 4.198650993474429809884798868723112488797 \cdot 10^{74}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\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 r26090 = b_2;
        double r26091 = -r26090;
        double r26092 = r26090 * r26090;
        double r26093 = a;
        double r26094 = c;
        double r26095 = r26093 * r26094;
        double r26096 = r26092 - r26095;
        double r26097 = sqrt(r26096);
        double r26098 = r26091 - r26097;
        double r26099 = r26098 / r26093;
        return r26099;
}
↓
double f(double a, double b_2, double c) {
        double r26100 = b_2;
        double r26101 = -3.7277239731487426e+105;
        bool r26102 = r26100 <= r26101;
        double r26103 = -0.5;
        double r26104 = c;
        double r26105 = r26104 / r26100;
        double r26106 = r26103 * r26105;
        double r26107 = 1.3645292833571162e-246;
        bool r26108 = r26100 <= r26107;
        double r26109 = r26100 * r26100;
        double r26110 = a;
        double r26111 = r26110 * r26104;
        double r26112 = r26109 - r26111;
        double r26113 = sqrt(r26112);
        double r26114 = r26113 - r26100;
        double r26115 = r26104 / r26114;
        double r26116 = 4.19865099347443e+74;
        bool r26117 = r26100 <= r26116;
        double r26118 = -r26100;
        double r26119 = r26118 / r26110;
        double r26120 = r26113 / r26110;
        double r26121 = r26119 - r26120;
        double r26122 = 0.5;
        double r26123 = r26122 * r26105;
        double r26124 = 2.0;
        double r26125 = r26100 / r26110;
        double r26126 = r26124 * r26125;
        double r26127 = r26123 - r26126;
        double r26128 = r26117 ? r26121 : r26127;
        double r26129 = r26108 ? r26115 : r26128;
        double r26130 = r26102 ? r26106 : r26129;
        return r26130;
}

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 < -3.7277239731487426e+105

    1. Initial program 59.9

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

    2. Taylor expanded around -inf 2.4

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

    if -3.7277239731487426e+105 < b_2 < 1.3645292833571162e-246

    1. Initial program 30.7

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

    3. Applied flip--30.8

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

    4. Simplified15.7

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

    5. Simplified15.7

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

    7. Applied *-un-lft-identity15.7

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

    8. Applied *-un-lft-identity15.7

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

    9. Applied times-frac15.7

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

    10. Simplified15.7

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

    11. Simplified8.5

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

    if 1.3645292833571162e-246 < b_2 < 4.19865099347443e+74

    1. Initial program 9.3

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

    3. Applied div-sub9.3

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

    if 4.19865099347443e+74 < b_2

    1. Initial program 41.8

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

    2. Taylor expanded around inf 5.3

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.727723973148742563324153150500151191249 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.364529283357116165462040672443911208369 \cdot 10^{-246}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.198650993474429809884798868723112488797 \cdot 10^{74}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))