Average Error: 34.5 → 8.7
Time: 7.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.95460202615535102 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.82564652270824723 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 2.039304743620259 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.95460202615535102 \cdot 10^{-16}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.82564652270824723 \cdot 10^{-308}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 2.039304743620259 \cdot 10^{111}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r30034 = b_2;
        double r30035 = -r30034;
        double r30036 = r30034 * r30034;
        double r30037 = a;
        double r30038 = c;
        double r30039 = r30037 * r30038;
        double r30040 = r30036 - r30039;
        double r30041 = sqrt(r30040);
        double r30042 = r30035 - r30041;
        double r30043 = r30042 / r30037;
        return r30043;
}
↓
double f(double a, double b_2, double c) {
        double r30044 = b_2;
        double r30045 = -1.954602026155351e-16;
        bool r30046 = r30044 <= r30045;
        double r30047 = -0.5;
        double r30048 = c;
        double r30049 = r30048 / r30044;
        double r30050 = r30047 * r30049;
        double r30051 = -3.825646522708247e-308;
        bool r30052 = r30044 <= r30051;
        double r30053 = 1.0;
        double r30054 = r30044 * r30044;
        double r30055 = a;
        double r30056 = r30055 * r30048;
        double r30057 = r30054 - r30056;
        double r30058 = sqrt(r30057);
        double r30059 = r30058 - r30044;
        double r30060 = r30059 / r30055;
        double r30061 = r30060 / r30048;
        double r30062 = r30053 / r30061;
        double r30063 = r30062 / r30055;
        double r30064 = 2.0393047436202585e+111;
        bool r30065 = r30044 <= r30064;
        double r30066 = -r30044;
        double r30067 = r30066 - r30058;
        double r30068 = r30053 / r30055;
        double r30069 = r30067 * r30068;
        double r30070 = -2.0;
        double r30071 = r30044 / r30055;
        double r30072 = r30070 * r30071;
        double r30073 = r30065 ? r30069 : r30072;
        double r30074 = r30052 ? r30063 : r30073;
        double r30075 = r30046 ? r30050 : r30074;
        return r30075;
}

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 < -1.954602026155351e-16

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 6.7

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

    if -1.954602026155351e-16 < b_2 < -3.825646522708247e-308

    1. Initial program 25.3

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

    3. Applied flip--25.3

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

    4. Simplified17.6

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

    5. Simplified17.6

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

    7. Applied clear-num17.6

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

    8. Simplified14.2

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

    if -3.825646522708247e-308 < b_2 < 2.0393047436202585e+111

    1. Initial program 9.7

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

    3. Applied div-inv9.9

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

    if 2.0393047436202585e+111 < b_2

    1. Initial program 49.6

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

    3. Applied flip--63.2

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    7. Applied div-inv62.2

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

    8. Taylor expanded around 0 3.7

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.95460202615535102 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.82564652270824723 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 2.039304743620259 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))