Average Error: 34.2 → 10.0
Time: 17.1s
Precision: 64
$b \cdot b - \left(4 \cdot a\right) \cdot c \ge 0.0$
$\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}$
$\begin{array}{l} \mathbf{if}\;b \le -2.940001550635406357572527191036557327772 \cdot 10^{93}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.592868408155405548302080990651854167189 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}$
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
\begin{array}{l}
\mathbf{if}\;b \le -2.940001550635406357572527191036557327772 \cdot 10^{93}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.592868408155405548302080990651854167189 \cdot 10^{-64}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{a \cdot 2}\\

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

\end{array}
double f(double b, double a, double c) {
double r23805906 = b;
double r23805907 = -r23805906;
double r23805908 = r23805906 * r23805906;
double r23805909 = 4.0;
double r23805910 = a;
double r23805911 = r23805909 * r23805910;
double r23805912 = c;
double r23805913 = r23805911 * r23805912;
double r23805914 = r23805908 - r23805913;
double r23805915 = sqrt(r23805914);
double r23805916 = r23805907 + r23805915;
double r23805917 = 2.0;
double r23805918 = r23805917 * r23805910;
double r23805919 = r23805916 / r23805918;
return r23805919;
}

double f(double b, double a, double c) {
double r23805920 = b;
double r23805921 = -2.9400015506354064e+93;
bool r23805922 = r23805920 <= r23805921;
double r23805923 = c;
double r23805924 = r23805923 / r23805920;
double r23805925 = a;
double r23805926 = r23805920 / r23805925;
double r23805927 = r23805924 - r23805926;
double r23805928 = 1.0;
double r23805929 = r23805927 * r23805928;
double r23805930 = 2.5928684081554055e-64;
bool r23805931 = r23805920 <= r23805930;
double r23805932 = r23805920 * r23805920;
double r23805933 = 4.0;
double r23805934 = r23805925 * r23805933;
double r23805935 = r23805934 * r23805923;
double r23805936 = r23805932 - r23805935;
double r23805937 = sqrt(r23805936);
double r23805938 = r23805937 - r23805920;
double r23805939 = 1.0;
double r23805940 = 2.0;
double r23805941 = r23805925 * r23805940;
double r23805942 = r23805939 / r23805941;
double r23805943 = r23805938 * r23805942;
double r23805944 = -1.0;
double r23805945 = r23805924 * r23805944;
double r23805946 = r23805931 ? r23805943 : r23805945;
double r23805947 = r23805922 ? r23805929 : r23805946;
return r23805947;
}

Try it out

Results

 In Out
Enter valid numbers for all inputs

Derivation

1. Split input into 3 regimes
2. if b < -2.9400015506354064e+93

1. Initial program 45.8

$\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}$
2. Simplified45.8

$\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}}$
3. Taylor expanded around -inf 3.3

$\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}$
4. Simplified3.3

$\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1}$

if -2.9400015506354064e+93 < b < 2.5928684081554055e-64

1. Initial program 13.0

$\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}$
2. Simplified13.0

$\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}}$
3. Using strategy rm
4. Applied clear-num13.1

$\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}$
5. Using strategy rm
6. Applied div-inv13.2

$\leadsto \frac{1}{\color{blue}{\left(a \cdot 2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}$
7. Applied *-un-lft-identity13.2

$\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(a \cdot 2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}$
8. Applied times-frac13.2

$\leadsto \color{blue}{\frac{1}{a \cdot 2} \cdot \frac{1}{\frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}$
9. Simplified13.1

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

if 2.5928684081554055e-64 < b

1. Initial program 53.6

$\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}$
2. Simplified53.6

$\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}}$
3. Taylor expanded around inf 9.2

$\leadsto \color{blue}{-1 \cdot \frac{c}{b}}$
3. Recombined 3 regimes into one program.
4. Final simplification10.0

$\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.940001550635406357572527191036557327772 \cdot 10^{93}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.592868408155405548302080990651854167189 \cdot 10^{-64}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}$

Reproduce

herbie shell --seed 1
(FPCore (b a c)
:name "((-b)+sqrt(b*b-4*a*c))/(2*a)"
:pre (>= (- (* b b) (* (* 4.0 a) c)) 0.0)
(/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))