Average Error: 34.1 → 10.0
Time: 14.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{\left(-c\right) \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\
\;\;\;\;\frac{\frac{\left(-c\right) \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\

\mathbf{elif}\;b_2 \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r26174 = b_2;
        double r26175 = -r26174;
        double r26176 = r26174 * r26174;
        double r26177 = a;
        double r26178 = c;
        double r26179 = r26177 * r26178;
        double r26180 = r26176 - r26179;
        double r26181 = sqrt(r26180);
        double r26182 = r26175 - r26181;
        double r26183 = r26182 / r26177;
        return r26183;
}
↓
double f(double a, double b_2, double c) {
        double r26184 = b_2;
        double r26185 = -5.84038254482515e+46;
        bool r26186 = r26184 <= r26185;
        double r26187 = -0.5;
        double r26188 = c;
        double r26189 = r26188 / r26184;
        double r26190 = r26187 * r26189;
        double r26191 = -7.877985662156599e-94;
        bool r26192 = r26184 <= r26191;
        double r26193 = -r26188;
        double r26194 = a;
        double r26195 = r26193 * r26194;
        double r26196 = r26184 * r26184;
        double r26197 = r26188 * r26194;
        double r26198 = r26196 - r26197;
        double r26199 = sqrt(r26198);
        double r26200 = r26184 - r26199;
        double r26201 = r26195 / r26200;
        double r26202 = r26201 / r26194;
        double r26203 = -6.596302400897662e-136;
        bool r26204 = r26184 <= r26203;
        double r26205 = 7.501979458872916e+77;
        bool r26206 = r26184 <= r26205;
        double r26207 = 1.0;
        double r26208 = -r26184;
        double r26209 = r26208 - r26199;
        double r26210 = r26194 / r26209;
        double r26211 = r26207 / r26210;
        double r26212 = -2.0;
        double r26213 = r26184 / r26194;
        double r26214 = r26212 * r26213;
        double r26215 = r26206 ? r26211 : r26214;
        double r26216 = r26204 ? r26190 : r26215;
        double r26217 = r26192 ? r26202 : r26216;
        double r26218 = r26186 ? r26190 : r26217;
        return r26218;
}

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 < -5.84038254482515e+46 or -7.877985662156599e-94 < b_2 < -6.596302400897662e-136

    1. Initial program 54.3

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

    2. Simplified54.3

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

    3. Taylor expanded around -inf 8.3

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

    if -5.84038254482515e+46 < b_2 < -7.877985662156599e-94

    1. Initial program 40.2

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

    2. Simplified40.2

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

    4. Applied flip-+40.2

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

    5. Simplified15.2

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

    if -6.596302400897662e-136 < b_2 < 7.501979458872916e+77

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied clear-num12.1

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

    5. Simplified12.1

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

    if 7.501979458872916e+77 < b_2

    1. Initial program 42.5

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

    2. Simplified42.5

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

    4. Applied clear-num42.6

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

    5. Simplified42.6

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

    6. Taylor expanded around 0 5.1

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

    7. Simplified5.1

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{\left(-c\right) \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))