Average Error: 34.3 → 8.2
Time: 5.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.3726940620353851 \cdot 10^{45}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.37755736165955163 \cdot 10^{-277}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{a} \cdot \frac{\sqrt[3]{a}}{c}\right) \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.9740449679534498 \cdot 10^{93}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.3726940620353851 \cdot 10^{45}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.37755736165955163 \cdot 10^{-277}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{a} \cdot \frac{\sqrt[3]{a}}{c}\right) \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.9740449679534498 \cdot 10^{93}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r18540 = b_2;
        double r18541 = -r18540;
        double r18542 = r18540 * r18540;
        double r18543 = a;
        double r18544 = c;
        double r18545 = r18543 * r18544;
        double r18546 = r18542 - r18545;
        double r18547 = sqrt(r18546);
        double r18548 = r18541 - r18547;
        double r18549 = r18548 / r18543;
        return r18549;
}
↓
double f(double a, double b_2, double c) {
        double r18550 = b_2;
        double r18551 = -1.3726940620353851e+45;
        bool r18552 = r18550 <= r18551;
        double r18553 = -0.5;
        double r18554 = c;
        double r18555 = r18554 / r18550;
        double r18556 = r18553 * r18555;
        double r18557 = -1.3775573616595516e-277;
        bool r18558 = r18550 <= r18557;
        double r18559 = 1.0;
        double r18560 = r18550 * r18550;
        double r18561 = a;
        double r18562 = r18561 * r18554;
        double r18563 = r18560 - r18562;
        double r18564 = sqrt(r18563);
        double r18565 = r18564 - r18550;
        double r18566 = sqrt(r18565);
        double r18567 = r18559 / r18566;
        double r18568 = cbrt(r18561);
        double r18569 = r18568 * r18568;
        double r18570 = r18569 / r18561;
        double r18571 = r18568 / r18554;
        double r18572 = r18570 * r18571;
        double r18573 = r18572 * r18566;
        double r18574 = r18567 / r18573;
        double r18575 = 1.9740449679534498e+93;
        bool r18576 = r18550 <= r18575;
        double r18577 = -r18550;
        double r18578 = r18577 - r18564;
        double r18579 = r18559 / r18561;
        double r18580 = r18578 * r18579;
        double r18581 = 0.5;
        double r18582 = r18581 * r18555;
        double r18583 = 2.0;
        double r18584 = r18550 / r18561;
        double r18585 = r18583 * r18584;
        double r18586 = r18582 - r18585;
        double r18587 = r18576 ? r18580 : r18586;
        double r18588 = r18558 ? r18574 : r18587;
        double r18589 = r18552 ? r18556 : r18588;
        return r18589;
}

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 < -1.3726940620353851e+45

    1. Initial program 57.2

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

    2. Taylor expanded around -inf 4.0

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

    if -1.3726940620353851e+45 < b_2 < -1.3775573616595516e-277

    1. Initial program 30.8

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

    3. Applied flip--30.8

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

    4. Simplified16.7

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

    5. Simplified16.7

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

    7. Applied add-sqr-sqrt16.9

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

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

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

    9. Applied times-frac16.9

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

    10. Applied associate-/l*16.8

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

    11. Simplified16.7

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

    13. Applied add-cube-cbrt17.3

      \[\leadsto \frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{a \cdot c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    14. Applied times-frac14.5

      \[\leadsto \frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{a} \cdot \frac{\sqrt[3]{a}}{c}\right)} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    if -1.3775573616595516e-277 < b_2 < 1.9740449679534498e+93

    1. Initial program 9.2

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

    3. Applied div-inv9.4

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

    if 1.9740449679534498e+93 < b_2

    1. Initial program 46.1

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.3726940620353851 \cdot 10^{45}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.37755736165955163 \cdot 10^{-277}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{a} \cdot \frac{\sqrt[3]{a}}{c}\right) \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.9740449679534498 \cdot 10^{93}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))