Average Error: 33.6 → 10.0
Time: 19.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.4515142736560382 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.822071483797687 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{\frac{a}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.4515142736560382 \cdot 10^{+31}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.822071483797687 \cdot 10^{-95}:\\
\;\;\;\;\frac{a}{\frac{a}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\

\mathbf{elif}\;b_2 \le 0.17389787404847717:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r4243378 = b_2;
        double r4243379 = -r4243378;
        double r4243380 = r4243378 * r4243378;
        double r4243381 = a;
        double r4243382 = c;
        double r4243383 = r4243381 * r4243382;
        double r4243384 = r4243380 - r4243383;
        double r4243385 = sqrt(r4243384);
        double r4243386 = r4243379 - r4243385;
        double r4243387 = r4243386 / r4243381;
        return r4243387;
}
↓
double f(double a, double b_2, double c) {
        double r4243388 = b_2;
        double r4243389 = -2.4515142736560382e+31;
        bool r4243390 = r4243388 <= r4243389;
        double r4243391 = -0.5;
        double r4243392 = c;
        double r4243393 = r4243392 / r4243388;
        double r4243394 = r4243391 * r4243393;
        double r4243395 = -6.822071483797687e-95;
        bool r4243396 = r4243388 <= r4243395;
        double r4243397 = a;
        double r4243398 = r4243388 * r4243388;
        double r4243399 = r4243397 * r4243392;
        double r4243400 = r4243398 - r4243399;
        double r4243401 = sqrt(r4243400);
        double r4243402 = r4243401 - r4243388;
        double r4243403 = r4243392 / r4243402;
        double r4243404 = r4243397 / r4243403;
        double r4243405 = r4243397 / r4243404;
        double r4243406 = 0.17389787404847717;
        bool r4243407 = r4243388 <= r4243406;
        double r4243408 = -r4243388;
        double r4243409 = r4243408 - r4243401;
        double r4243410 = r4243409 / r4243397;
        double r4243411 = 0.5;
        double r4243412 = r4243393 * r4243411;
        double r4243413 = 2.0;
        double r4243414 = r4243388 / r4243397;
        double r4243415 = r4243413 * r4243414;
        double r4243416 = r4243412 - r4243415;
        double r4243417 = r4243407 ? r4243410 : r4243416;
        double r4243418 = r4243396 ? r4243405 : r4243417;
        double r4243419 = r4243390 ? r4243394 : r4243418;
        return r4243419;
}

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 < -2.4515142736560382e+31

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 4.3

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

    if -2.4515142736560382e+31 < b_2 < -6.822071483797687e-95

    1. Initial program 39.5

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

    3. Applied flip--39.5

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

    4. Simplified14.0

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

    5. Simplified14.0

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

    7. Applied *-un-lft-identity14.0

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

    8. Applied times-frac11.4

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

    9. Applied associate-/l*19.9

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

    if -6.822071483797687e-95 < b_2 < 0.17389787404847717

    1. Initial program 14.2

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

    if 0.17389787404847717 < b_2

    1. Initial program 29.8

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

    2. Taylor expanded around inf 7.3

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.4515142736560382 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.822071483797687 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{\frac{a}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))