Average Error: 34.9 → 6.9
Time: 9.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.039316568980459063820204785769574103851 \cdot 10^{69}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le \frac{-3742251048940711}{2.083754510274954460649953684600393061608 \cdot 10^{239}}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 3.120866175169106465948772460927385236705 \cdot 10^{88}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.039316568980459063820204785769574103851 \cdot 10^{69}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le \frac{-3742251048940711}{2.083754510274954460649953684600393061608 \cdot 10^{239}}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 3.120866175169106465948772460927385236705 \cdot 10^{88}:\\
\;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r26254 = b_2;
        double r26255 = -r26254;
        double r26256 = r26254 * r26254;
        double r26257 = a;
        double r26258 = c;
        double r26259 = r26257 * r26258;
        double r26260 = r26256 - r26259;
        double r26261 = sqrt(r26260);
        double r26262 = r26255 + r26261;
        double r26263 = r26262 / r26257;
        return r26263;
}
↓
double f(double a, double b_2, double c) {
        double r26264 = b_2;
        double r26265 = -1.039316568980459e+69;
        bool r26266 = r26264 <= r26265;
        double r26267 = 0.5;
        double r26268 = c;
        double r26269 = r26268 / r26264;
        double r26270 = r26267 * r26269;
        double r26271 = 2.0;
        double r26272 = a;
        double r26273 = r26264 / r26272;
        double r26274 = r26271 * r26273;
        double r26275 = r26270 - r26274;
        double r26276 = -3742251048940711.0;
        double r26277 = 2.0837545102749545e+239;
        double r26278 = r26276 / r26277;
        bool r26279 = r26264 <= r26278;
        double r26280 = 1.0;
        double r26281 = r26264 * r26264;
        double r26282 = r26272 * r26268;
        double r26283 = r26281 - r26282;
        double r26284 = sqrt(r26283);
        double r26285 = r26284 - r26264;
        double r26286 = r26272 / r26285;
        double r26287 = r26280 / r26286;
        double r26288 = 3.1208661751691065e+88;
        bool r26289 = r26264 <= r26288;
        double r26290 = r26280 / r26268;
        double r26291 = -r26264;
        double r26292 = r26291 - r26284;
        double r26293 = r26290 * r26292;
        double r26294 = r26280 / r26293;
        double r26295 = -0.5;
        double r26296 = r26295 * r26269;
        double r26297 = r26289 ? r26294 : r26296;
        double r26298 = r26279 ? r26287 : r26297;
        double r26299 = r26266 ? r26275 : r26298;
        return r26299;
}

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 < -1.039316568980459e+69

    1. Initial program 40.9

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

    2. Taylor expanded around -inf 4.8

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

    if -1.039316568980459e+69 < b_2 < -1.7959174319660695e-224

    1. Initial program 8.1

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

    3. Applied clear-num8.2

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

    4. Simplified8.2

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

    if -1.7959174319660695e-224 < b_2 < 3.1208661751691065e+88

    1. Initial program 29.9

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

    3. Applied flip-+30.0

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

    4. Simplified16.8

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

    6. Applied *-un-lft-identity16.8

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

    7. Applied times-frac14.4

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

    8. Simplified14.4

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

    10. Applied clear-num14.6

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

    11. Simplified10.2

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

    if 3.1208661751691065e+88 < b_2

    1. Initial program 59.5

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.039316568980459063820204785769574103851 \cdot 10^{69}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le \frac{-3742251048940711}{2.083754510274954460649953684600393061608 \cdot 10^{239}}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 3.120866175169106465948772460927385236705 \cdot 10^{88}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))