Average Error: 33.8 → 9.7
Time: 14.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 2.0569776426586135 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.93152454634661985 \cdot 10^{153}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 2.0569776426586135 \cdot 10^{-106}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r8976 = b_2;
        double r8977 = -r8976;
        double r8978 = r8976 * r8976;
        double r8979 = a;
        double r8980 = c;
        double r8981 = r8979 * r8980;
        double r8982 = r8978 - r8981;
        double r8983 = sqrt(r8982);
        double r8984 = r8977 + r8983;
        double r8985 = r8984 / r8979;
        return r8985;
}
↓
double f(double a, double b_2, double c) {
        double r8986 = b_2;
        double r8987 = -7.93152454634662e+153;
        bool r8988 = r8986 <= r8987;
        double r8989 = 0.5;
        double r8990 = c;
        double r8991 = r8990 / r8986;
        double r8992 = r8989 * r8991;
        double r8993 = 2.0;
        double r8994 = a;
        double r8995 = r8986 / r8994;
        double r8996 = r8993 * r8995;
        double r8997 = r8992 - r8996;
        double r8998 = 2.0569776426586135e-106;
        bool r8999 = r8986 <= r8998;
        double r9000 = r8986 * r8986;
        double r9001 = r8994 * r8990;
        double r9002 = r9000 - r9001;
        double r9003 = sqrt(r9002);
        double r9004 = r9003 - r8986;
        double r9005 = r9004 / r8994;
        double r9006 = -0.5;
        double r9007 = r9006 * r8991;
        double r9008 = r8999 ? r9005 : r9007;
        double r9009 = r8988 ? r8997 : r9008;
        return r9009;
}

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 3 regimes

  2. if b_2 < -7.93152454634662e+153

    1. Initial program 63.8

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

    2. Simplified63.8

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

    3. Taylor expanded around -inf 1.9

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

    if -7.93152454634662e+153 < b_2 < 2.0569776426586135e-106

    1. Initial program 11.1

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

    2. Simplified11.1

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

    if 2.0569776426586135e-106 < b_2

    1. Initial program 52.0

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

    2. Simplified52.0

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

    3. Taylor expanded around inf 10.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 2.0569776426586135 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))