Average Error: 34.0 → 6.7
Time: 5.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.540948762724586449509800928955156519735 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.992191814191663114066128965938211373155 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{a}}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 5.41682251593061203594104438490271250843 \cdot 10^{93}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.540948762724586449509800928955156519735 \cdot 10^{153}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.992191814191663114066128965938211373155 \cdot 10^{-305}:\\
\;\;\;\;\frac{1}{\frac{\frac{a}{a}}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\

\mathbf{elif}\;b_2 \le 5.41682251593061203594104438490271250843 \cdot 10^{93}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17090 = b_2;
        double r17091 = -r17090;
        double r17092 = r17090 * r17090;
        double r17093 = a;
        double r17094 = c;
        double r17095 = r17093 * r17094;
        double r17096 = r17092 - r17095;
        double r17097 = sqrt(r17096);
        double r17098 = r17091 - r17097;
        double r17099 = r17098 / r17093;
        return r17099;
}
↓
double f(double a, double b_2, double c) {
        double r17100 = b_2;
        double r17101 = -6.5409487627245864e+153;
        bool r17102 = r17100 <= r17101;
        double r17103 = -0.5;
        double r17104 = c;
        double r17105 = r17104 / r17100;
        double r17106 = r17103 * r17105;
        double r17107 = -1.992191814191663e-305;
        bool r17108 = r17100 <= r17107;
        double r17109 = 1.0;
        double r17110 = a;
        double r17111 = r17110 / r17110;
        double r17112 = r17111 / r17104;
        double r17113 = r17100 * r17100;
        double r17114 = r17110 * r17104;
        double r17115 = r17113 - r17114;
        double r17116 = sqrt(r17115);
        double r17117 = r17116 - r17100;
        double r17118 = r17112 * r17117;
        double r17119 = r17109 / r17118;
        double r17120 = 5.416822515930612e+93;
        bool r17121 = r17100 <= r17120;
        double r17122 = -r17100;
        double r17123 = r17122 - r17116;
        double r17124 = r17110 / r17123;
        double r17125 = r17109 / r17124;
        double r17126 = -2.0;
        double r17127 = r17100 / r17110;
        double r17128 = r17126 * r17127;
        double r17129 = r17121 ? r17125 : r17128;
        double r17130 = r17108 ? r17119 : r17129;
        double r17131 = r17102 ? r17106 : r17130;
        return r17131;
}

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 < -6.5409487627245864e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 1.8

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

    if -6.5409487627245864e+153 < b_2 < -1.992191814191663e-305

    1. Initial program 34.9

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

    3. Applied clear-num35.0

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

    5. Applied add-exp-log40.5

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

    7. Applied flip--40.5

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

    8. Simplified18.5

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

    9. Simplified16.0

      \[\leadsto \frac{1}{\frac{a}{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}\]
    10. Using strategy rm

    11. Applied associate-/r/14.4

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

    12. Simplified8.4

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

    if -1.992191814191663e-305 < b_2 < 5.416822515930612e+93

    1. Initial program 9.1

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

    3. Applied clear-num9.3

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

    if 5.416822515930612e+93 < b_2

    1. Initial program 45.2

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

    3. Applied clear-num45.3

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

    4. Taylor expanded around 0 3.4

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.540948762724586449509800928955156519735 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.992191814191663114066128965938211373155 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{a}}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 5.41682251593061203594104438490271250843 \cdot 10^{93}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))