Average Error: 33.8 → 6.6
Time: 5.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.48395925749684231185755183204764945523 \cdot 10^{145}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.027416821538068536725790738835144506948 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}{c}}\\ \mathbf{elif}\;b_2 \le 4.397833618396295559623484253976393644285 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.48395925749684231185755183204764945523 \cdot 10^{145}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.027416821538068536725790738835144506948 \cdot 10^{-173}:\\
\;\;\;\;\frac{1}{\frac{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}{c}}\\

\mathbf{elif}\;b_2 \le 4.397833618396295559623484253976393644285 \cdot 10^{103}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r71354 = b_2;
        double r71355 = -r71354;
        double r71356 = r71354 * r71354;
        double r71357 = a;
        double r71358 = c;
        double r71359 = r71357 * r71358;
        double r71360 = r71356 - r71359;
        double r71361 = sqrt(r71360);
        double r71362 = r71355 - r71361;
        double r71363 = r71362 / r71357;
        return r71363;
}
↓
double f(double a, double b_2, double c) {
        double r71364 = b_2;
        double r71365 = -6.483959257496842e+145;
        bool r71366 = r71364 <= r71365;
        double r71367 = -0.5;
        double r71368 = c;
        double r71369 = r71368 / r71364;
        double r71370 = r71367 * r71369;
        double r71371 = 1.0274168215380685e-173;
        bool r71372 = r71364 <= r71371;
        double r71373 = 1.0;
        double r71374 = r71364 * r71364;
        double r71375 = a;
        double r71376 = r71375 * r71368;
        double r71377 = r71374 - r71376;
        double r71378 = sqrt(r71377);
        double r71379 = r71378 - r71364;
        double r71380 = r71373 * r71379;
        double r71381 = r71380 / r71368;
        double r71382 = r71373 / r71381;
        double r71383 = 4.3978336183962956e+103;
        bool r71384 = r71364 <= r71383;
        double r71385 = -r71364;
        double r71386 = r71385 - r71378;
        double r71387 = r71386 / r71375;
        double r71388 = -2.0;
        double r71389 = r71388 * r71364;
        double r71390 = r71389 / r71375;
        double r71391 = r71384 ? r71387 : r71390;
        double r71392 = r71372 ? r71382 : r71391;
        double r71393 = r71366 ? r71370 : r71392;
        return r71393;
}

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 < -6.483959257496842e+145

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 1.6

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

    if -6.483959257496842e+145 < b_2 < 1.0274168215380685e-173

    1. Initial program 30.5

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

    3. Applied flip--30.7

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

    4. Simplified15.4

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

    5. Simplified15.4

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

    7. Applied *-un-lft-identity15.4

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

    8. Applied *-un-lft-identity15.4

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

    9. Applied times-frac15.4

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

    10. Simplified15.4

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

    11. Simplified13.8

      \[\leadsto \frac{1 \cdot \color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
    12. Using strategy rm

    13. Applied clear-num13.8

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

    14. Simplified9.7

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

    if 1.0274168215380685e-173 < b_2 < 4.3978336183962956e+103

    1. Initial program 6.4

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

    if 4.3978336183962956e+103 < b_2

    1. Initial program 47.2

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

    3. Applied flip--63.4

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

    4. Simplified62.5

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

    5. Simplified62.5

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

    6. Taylor expanded around 0 4.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.48395925749684231185755183204764945523 \cdot 10^{145}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.027416821538068536725790738835144506948 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}{c}}\\ \mathbf{elif}\;b_2 \le 4.397833618396295559623484253976393644285 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))