Average Error: 34.1 → 8.5
Time: 18.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -128467017418558619648:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.43876958772637578815854414301947050583 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\ \mathbf{elif}\;b_2 \le 6.190496209585126281868131365991188760623 \cdot 10^{153}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -128467017418558619648:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -3.43876958772637578815854414301947050583 \cdot 10^{-134}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\

\mathbf{elif}\;b_2 \le 6.190496209585126281868131365991188760623 \cdot 10^{153}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r25254 = b_2;
        double r25255 = -r25254;
        double r25256 = r25254 * r25254;
        double r25257 = a;
        double r25258 = c;
        double r25259 = r25257 * r25258;
        double r25260 = r25256 - r25259;
        double r25261 = sqrt(r25260);
        double r25262 = r25255 + r25261;
        double r25263 = r25262 / r25257;
        return r25263;
}
↓
double f(double a, double b_2, double c) {
        double r25264 = b_2;
        double r25265 = -1.2846701741855862e+20;
        bool r25266 = r25264 <= r25265;
        double r25267 = 0.5;
        double r25268 = c;
        double r25269 = r25268 / r25264;
        double r25270 = r25267 * r25269;
        double r25271 = 2.0;
        double r25272 = a;
        double r25273 = r25264 / r25272;
        double r25274 = r25271 * r25273;
        double r25275 = r25270 - r25274;
        double r25276 = -3.438769587726376e-134;
        bool r25277 = r25264 <= r25276;
        double r25278 = r25272 * r25268;
        double r25279 = r25264 * r25264;
        double r25280 = r25279 - r25278;
        double r25281 = sqrt(r25280);
        double r25282 = r25281 - r25264;
        double r25283 = r25278 / r25282;
        double r25284 = r25278 / r25283;
        double r25285 = r25284 / r25272;
        double r25286 = 6.190496209585126e+153;
        bool r25287 = r25264 <= r25286;
        double r25288 = -r25264;
        double r25289 = r25288 - r25281;
        double r25290 = r25268 / r25289;
        double r25291 = -0.5;
        double r25292 = r25291 * r25269;
        double r25293 = r25287 ? r25290 : r25292;
        double r25294 = r25277 ? r25285 : r25293;
        double r25295 = r25266 ? r25275 : r25294;
        return r25295;
}

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 < -1.2846701741855862e+20

    1. Initial program 34.5

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

    2. Taylor expanded around -inf 6.0

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

    if -1.2846701741855862e+20 < b_2 < -3.438769587726376e-134

    1. Initial program 4.7

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

    3. Applied flip-+37.0

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

    4. Simplified37.1

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

    6. Applied flip--37.1

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

    7. Simplified16.0

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

    8. Simplified16.0

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

    if -3.438769587726376e-134 < b_2 < 6.190496209585126e+153

    1. Initial program 30.2

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

    3. Applied flip-+31.1

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

    4. Simplified16.9

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

    6. Applied *-un-lft-identity16.9

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

    7. Applied *-un-lft-identity16.9

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

    8. Applied times-frac16.9

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

    9. Simplified16.9

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

    10. Simplified10.5

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

    if 6.190496209585126e+153 < b_2

    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 1.5

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -128467017418558619648:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.43876958772637578815854414301947050583 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\ \mathbf{elif}\;b_2 \le 6.190496209585126281868131365991188760623 \cdot 10^{153}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))