Average Error: 34.4 → 8.2
Time: 7.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.2038615298990798 \cdot 10^{25}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.57911224209110995 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.06577896741184288 \cdot 10^{116}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.2038615298990798 \cdot 10^{25}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 9.57911224209110995 \cdot 10^{-234}:\\
\;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 4.06577896741184288 \cdot 10^{116}:\\
\;\;\;\;\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 r72593 = b_2;
        double r72594 = -r72593;
        double r72595 = r72593 * r72593;
        double r72596 = a;
        double r72597 = c;
        double r72598 = r72596 * r72597;
        double r72599 = r72595 - r72598;
        double r72600 = sqrt(r72599);
        double r72601 = r72594 - r72600;
        double r72602 = r72601 / r72596;
        return r72602;
}
↓
double f(double a, double b_2, double c) {
        double r72603 = b_2;
        double r72604 = -9.20386152989908e+25;
        bool r72605 = r72603 <= r72604;
        double r72606 = -0.5;
        double r72607 = c;
        double r72608 = r72607 / r72603;
        double r72609 = r72606 * r72608;
        double r72610 = 9.57911224209111e-234;
        bool r72611 = r72603 <= r72610;
        double r72612 = a;
        double r72613 = r72603 * r72603;
        double r72614 = r72612 * r72607;
        double r72615 = r72613 - r72614;
        double r72616 = sqrt(r72615);
        double r72617 = r72616 - r72603;
        double r72618 = r72617 / r72607;
        double r72619 = r72612 / r72618;
        double r72620 = r72619 / r72612;
        double r72621 = 4.065778967411843e+116;
        bool r72622 = r72603 <= r72621;
        double r72623 = -r72603;
        double r72624 = r72623 - r72616;
        double r72625 = 1.0;
        double r72626 = r72625 / r72612;
        double r72627 = r72624 * r72626;
        double r72628 = 0.5;
        double r72629 = r72628 * r72608;
        double r72630 = 2.0;
        double r72631 = r72603 / r72612;
        double r72632 = r72630 * r72631;
        double r72633 = r72629 - r72632;
        double r72634 = r72622 ? r72627 : r72633;
        double r72635 = r72611 ? r72620 : r72634;
        double r72636 = r72605 ? r72609 : r72635;
        return r72636;
}

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 < -9.20386152989908e+25

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 5.0

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

    if -9.20386152989908e+25 < b_2 < 9.57911224209111e-234

    1. Initial program 25.9

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

    3. Applied flip--26.0

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

    4. Simplified17.1

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

    5. Simplified17.1

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

    7. Applied *-un-lft-identity17.1

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

    8. Applied associate-/r*17.1

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

    9. Simplified14.3

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

    if 9.57911224209111e-234 < b_2 < 4.065778967411843e+116

    1. Initial program 7.8

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

    3. Applied div-inv7.9

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

    if 4.065778967411843e+116 < b_2

    1. Initial program 51.7

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.2038615298990798 \cdot 10^{25}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.57911224209110995 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.06577896741184288 \cdot 10^{116}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))