Average Error: 34.1 → 8.2
Time: 4.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.4834941205945284 \cdot 10^{67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.329606077024855 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a}{\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{elif}\;b_2 \le 4.8331091207749691 \cdot 10^{125}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.4834941205945284 \cdot 10^{67}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.329606077024855 \cdot 10^{-296}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{a}{\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\

\mathbf{elif}\;b_2 \le 4.8331091207749691 \cdot 10^{125}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{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 r87055 = b_2;
        double r87056 = -r87055;
        double r87057 = r87055 * r87055;
        double r87058 = a;
        double r87059 = c;
        double r87060 = r87058 * r87059;
        double r87061 = r87057 - r87060;
        double r87062 = sqrt(r87061);
        double r87063 = r87056 - r87062;
        double r87064 = r87063 / r87058;
        return r87064;
}
↓
double f(double a, double b_2, double c) {
        double r87065 = b_2;
        double r87066 = -2.4834941205945284e+67;
        bool r87067 = r87065 <= r87066;
        double r87068 = -0.5;
        double r87069 = c;
        double r87070 = r87069 / r87065;
        double r87071 = r87068 * r87070;
        double r87072 = 2.329606077024855e-296;
        bool r87073 = r87065 <= r87072;
        double r87074 = 1.0;
        double r87075 = a;
        double r87076 = r87074 / r87075;
        double r87077 = -r87065;
        double r87078 = r87065 * r87065;
        double r87079 = r87075 * r87069;
        double r87080 = r87078 - r87079;
        double r87081 = sqrt(r87080);
        double r87082 = r87077 + r87081;
        double r87083 = r87082 / r87069;
        double r87084 = r87075 / r87083;
        double r87085 = r87076 * r87084;
        double r87086 = 4.833109120774969e+125;
        bool r87087 = r87065 <= r87086;
        double r87088 = r87077 - r87081;
        double r87089 = r87088 / r87075;
        double r87090 = 0.5;
        double r87091 = r87090 * r87070;
        double r87092 = 2.0;
        double r87093 = r87065 / r87075;
        double r87094 = r87092 * r87093;
        double r87095 = r87091 - r87094;
        double r87096 = r87087 ? r87089 : r87095;
        double r87097 = r87073 ? r87085 : r87096;
        double r87098 = r87067 ? r87071 : r87097;
        return r87098;
}

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 < -2.4834941205945284e+67

    1. Initial program 57.7

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

    2. Taylor expanded around -inf 3.7

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

    if -2.4834941205945284e+67 < b_2 < 2.329606077024855e-296

    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 clear-num30.6

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

    5. Applied div-inv30.6

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

    6. Applied add-cube-cbrt30.6

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

    7. Applied times-frac30.6

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

    8. Simplified30.6

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

    9. Simplified30.6

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

    11. Applied flip--30.6

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

    12. Applied frac-times35.6

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

    13. Simplified22.4

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

    15. Applied *-un-lft-identity22.4

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

    16. Applied times-frac17.5

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

    17. Simplified15.1

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

    if 2.329606077024855e-296 < b_2 < 4.833109120774969e+125

    1. Initial program 8.5

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

    if 4.833109120774969e+125 < b_2

    1. Initial program 53.6

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.4834941205945284 \cdot 10^{67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.329606077024855 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a}{\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{elif}\;b_2 \le 4.8331091207749691 \cdot 10^{125}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))