Average Error: 34.2 → 10.5
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r102997 = b;
        double r102998 = -r102997;
        double r102999 = r102997 * r102997;
        double r103000 = 4.0;
        double r103001 = a;
        double r103002 = c;
        double r103003 = r103001 * r103002;
        double r103004 = r103000 * r103003;
        double r103005 = r102999 - r103004;
        double r103006 = sqrt(r103005);
        double r103007 = r102998 - r103006;
        double r103008 = 2.0;
        double r103009 = r103008 * r103001;
        double r103010 = r103007 / r103009;
        return r103010;
}
↓
double f(double a, double b, double c) {
        double r103011 = b;
        double r103012 = -2.763265898390889e-72;
        bool r103013 = r103011 <= r103012;
        double r103014 = -1.0;
        double r103015 = c;
        double r103016 = r103015 / r103011;
        double r103017 = r103014 * r103016;
        double r103018 = 6.568668442325333e+48;
        bool r103019 = r103011 <= r103018;
        double r103020 = -r103011;
        double r103021 = r103011 * r103011;
        double r103022 = 4.0;
        double r103023 = a;
        double r103024 = r103023 * r103015;
        double r103025 = r103022 * r103024;
        double r103026 = r103021 - r103025;
        double r103027 = sqrt(r103026);
        double r103028 = r103020 - r103027;
        double r103029 = 1.0;
        double r103030 = 2.0;
        double r103031 = r103030 * r103023;
        double r103032 = r103029 / r103031;
        double r103033 = r103028 * r103032;
        double r103034 = 1.0;
        double r103035 = r103011 / r103023;
        double r103036 = r103016 - r103035;
        double r103037 = r103034 * r103036;
        double r103038 = r103019 ? r103033 : r103037;
        double r103039 = r103013 ? r103017 : r103038;
        return r103039;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.2
Target21.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.763265898390889e-72

    1. Initial program 52.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 9.2

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

    if -2.763265898390889e-72 < b < 6.568668442325333e+48

    1. Initial program 14.1

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv14.2

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 6.568668442325333e+48 < b

    1. Initial program 39.1

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 5.4

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

    3. Simplified5.4

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))