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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}{2}\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

\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 -3.12428337420519208 \cdot 10^{57}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}{2}\\

\mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}\\

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

\end{array}

double f(double a, double b, double c) {
        double r84147 = b;
        double r84148 = -r84147;
        double r84149 = r84147 * r84147;
        double r84150 = 4.0;
        double r84151 = a;
        double r84152 = c;
        double r84153 = r84151 * r84152;
        double r84154 = r84150 * r84153;
        double r84155 = r84149 - r84154;
        double r84156 = sqrt(r84155);
        double r84157 = r84148 + r84156;
        double r84158 = 2.0;
        double r84159 = r84158 * r84151;
        double r84160 = r84157 / r84159;
        return r84160;
}

↓

double f(double a, double b, double c) {
        double r84161 = b;
        double r84162 = -3.124283374205192e+57;
        bool r84163 = r84161 <= r84162;
        double r84164 = 2.0;
        double r84165 = c;
        double r84166 = r84165 / r84161;
        double r84167 = -2.0;
        double r84168 = r84161 * r84167;
        double r84169 = a;
        double r84170 = r84168 / r84169;
        double r84171 = fma(r84164, r84166, r84170);
        double r84172 = r84171 / r84164;
        double r84173 = 3.84613441880261e-81;
        bool r84174 = r84161 <= r84173;
        double r84175 = r84161 * r84161;
        double r84176 = 4.0;
        double r84177 = r84169 * r84165;
        double r84178 = r84176 * r84177;
        double r84179 = r84175 - r84178;
        double r84180 = sqrt(r84179);
        double r84181 = r84180 - r84161;
        double r84182 = r84181 / r84164;
        double r84183 = 1.0;
        double r84184 = r84183 / r84169;
        double r84185 = r84182 * r84184;
        double r84186 = -2.0;
        double r84187 = r84186 * r84166;
        double r84188 = r84187 / r84164;
        double r84189 = r84174 ? r84185 : r84188;
        double r84190 = r84163 ? r84172 : r84189;
        return r84190;
}

Original	33.8
Target	20.4
Herbie	10.1

Derivation

Split input into 3 regimes
if b < -3.124283374205192e+57
1. Initial program 39.5
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Simplified39.5
  \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}}\]
3. Using strategy rm
4. Applied div-inv39.6
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}}\]
5. Using strategy rm
6. Applied *-un-lft-identity39.6
  \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}\right)} \cdot \frac{1}{a}\]
7. Applied associate-*l*39.6
  \[\leadsto \color{blue}{1 \cdot \left(\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}\right)}\]
8. Simplified39.5
  \[\leadsto 1 \cdot \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}}\]
9. Taylor expanded around -inf 5.4
  \[\leadsto 1 \cdot \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]
10. Simplified5.4
  \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}}{2}\]
if -3.124283374205192e+57 < b < 3.84613441880261e-81
1. Initial program 12.7
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Simplified12.7
  \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}}\]
3. Using strategy rm
4. Applied div-inv12.8
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}}\]
if 3.84613441880261e-81 < b
1. Initial program 53.0
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Simplified53.0
  \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}}\]
3. Using strategy rm
4. Applied div-inv53.0
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}}\]
5. Using strategy rm
6. Applied *-un-lft-identity53.0
  \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}\right)} \cdot \frac{1}{a}\]
7. Applied associate-*l*53.0
  \[\leadsto \color{blue}{1 \cdot \left(\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}\right)}\]
8. Simplified53.0
  \[\leadsto 1 \cdot \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a}}{2}}\]
9. Taylor expanded around inf 9.5
  \[\leadsto 1 \cdot \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
Recombined 3 regimes into one program.
Final simplification10.1
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b \cdot -2}{a}\right)}{2}\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if b < -3.124283374205192e+57`

`if -3.124283374205192e+57 < b < 3.84613441880261e-81`

`if 3.84613441880261e-81 < b`

Reproduce