Average Error: 34.6 → 8.9
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.69515204409029045 \cdot 10^{47}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le -6.7344880337585643 \cdot 10^{-149}:\\ \;\;\;\;1 \cdot \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.5329642823852832 \cdot 10^{69}:\\ \;\;\;\;1 \cdot \left(\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.69515204409029045 \cdot 10^{47}:\\
\;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le -6.7344880337585643 \cdot 10^{-149}:\\
\;\;\;\;1 \cdot \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 2.5329642823852832 \cdot 10^{69}:\\
\;\;\;\;1 \cdot \left(\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r15185 = b_2;
        double r15186 = -r15185;
        double r15187 = r15185 * r15185;
        double r15188 = a;
        double r15189 = c;
        double r15190 = r15188 * r15189;
        double r15191 = r15187 - r15190;
        double r15192 = sqrt(r15191);
        double r15193 = r15186 - r15192;
        double r15194 = r15193 / r15188;
        return r15194;
}
↓
double f(double a, double b_2, double c) {
        double r15195 = b_2;
        double r15196 = -1.6951520440902905e+47;
        bool r15197 = r15195 <= r15196;
        double r15198 = 1.0;
        double r15199 = -0.5;
        double r15200 = c;
        double r15201 = r15200 / r15195;
        double r15202 = r15199 * r15201;
        double r15203 = r15198 * r15202;
        double r15204 = -6.734488033758564e-149;
        bool r15205 = r15195 <= r15204;
        double r15206 = 0.0;
        double r15207 = a;
        double r15208 = r15207 * r15200;
        double r15209 = r15206 + r15208;
        double r15210 = r15195 * r15195;
        double r15211 = r15210 - r15208;
        double r15212 = sqrt(r15211);
        double r15213 = r15212 - r15195;
        double r15214 = r15209 / r15213;
        double r15215 = r15214 / r15207;
        double r15216 = r15198 * r15215;
        double r15217 = 2.5329642823852832e+69;
        bool r15218 = r15195 <= r15217;
        double r15219 = -r15195;
        double r15220 = r15219 / r15207;
        double r15221 = r15212 / r15207;
        double r15222 = r15220 - r15221;
        double r15223 = r15198 * r15222;
        double r15224 = 0.5;
        double r15225 = r15224 * r15201;
        double r15226 = 2.0;
        double r15227 = r15195 / r15207;
        double r15228 = r15226 * r15227;
        double r15229 = r15225 - r15228;
        double r15230 = r15198 * r15229;
        double r15231 = r15218 ? r15223 : r15230;
        double r15232 = r15205 ? r15216 : r15231;
        double r15233 = r15197 ? r15203 : r15232;
        return r15233;
}

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 < -1.6951520440902905e+47

    1. Initial program 56.8

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

    3. Applied *-un-lft-identity56.8

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

    4. Applied *-un-lft-identity56.8

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

    5. Applied times-frac56.8

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

    6. Simplified56.8

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

    8. Applied div-sub58.0

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

    9. Taylor expanded around -inf 4.2

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

    if -1.6951520440902905e+47 < b_2 < -6.734488033758564e-149

    1. Initial program 37.9

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

    3. Applied *-un-lft-identity37.9

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

    4. Applied *-un-lft-identity37.9

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

    5. Applied times-frac37.9

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

    6. Simplified37.9

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

    8. Applied flip--37.9

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

    9. Simplified17.1

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

    10. Simplified17.1

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

    if -6.734488033758564e-149 < b_2 < 2.5329642823852832e+69

    1. Initial program 11.4

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

    3. Applied *-un-lft-identity11.4

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

    4. Applied *-un-lft-identity11.4

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

    5. Applied times-frac11.4

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

    6. Simplified11.4

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

    8. Applied div-sub11.3

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

    if 2.5329642823852832e+69 < b_2

    1. Initial program 42.5

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

    3. Applied *-un-lft-identity42.5

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

    4. Applied *-un-lft-identity42.5

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

    5. Applied times-frac42.5

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

    6. Simplified42.5

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

    7. Taylor expanded around inf 4.7

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.69515204409029045 \cdot 10^{47}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le -6.7344880337585643 \cdot 10^{-149}:\\ \;\;\;\;1 \cdot \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.5329642823852832 \cdot 10^{69}:\\ \;\;\;\;1 \cdot \left(\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))