Average Error: 33.6 → 9.9
Time: 22.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.9356897939138002 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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 -8.1855168042470635 \cdot 10^{-53}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.9356897939138002 \cdot 10^{+147}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;-\frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r4731173 = b;
        double r4731174 = -r4731173;
        double r4731175 = r4731173 * r4731173;
        double r4731176 = 4.0;
        double r4731177 = a;
        double r4731178 = c;
        double r4731179 = r4731177 * r4731178;
        double r4731180 = r4731176 * r4731179;
        double r4731181 = r4731175 - r4731180;
        double r4731182 = sqrt(r4731181);
        double r4731183 = r4731174 - r4731182;
        double r4731184 = 2.0;
        double r4731185 = r4731184 * r4731177;
        double r4731186 = r4731183 / r4731185;
        return r4731186;
}
↓
double f(double a, double b, double c) {
        double r4731187 = b;
        double r4731188 = -8.1855168042470635e-53;
        bool r4731189 = r4731187 <= r4731188;
        double r4731190 = c;
        double r4731191 = r4731190 / r4731187;
        double r4731192 = -r4731191;
        double r4731193 = 2.9356897939138002e+147;
        bool r4731194 = r4731187 <= r4731193;
        double r4731195 = -r4731187;
        double r4731196 = r4731187 * r4731187;
        double r4731197 = a;
        double r4731198 = r4731190 * r4731197;
        double r4731199 = 4.0;
        double r4731200 = r4731198 * r4731199;
        double r4731201 = r4731196 - r4731200;
        double r4731202 = sqrt(r4731201);
        double r4731203 = r4731195 - r4731202;
        double r4731204 = 2.0;
        double r4731205 = r4731197 * r4731204;
        double r4731206 = r4731203 / r4731205;
        double r4731207 = r4731187 / r4731197;
        double r4731208 = -r4731207;
        double r4731209 = r4731194 ? r4731206 : r4731208;
        double r4731210 = r4731189 ? r4731192 : r4731209;
        return r4731210;
}

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

Original33.6
Target20.6
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.1855168042470635e-53

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 7.5

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

    3. Simplified7.5

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

    if -8.1855168042470635e-53 < b < 2.9356897939138002e+147

    1. Initial program 13.3

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

    if 2.9356897939138002e+147 < b

    1. Initial program 58.7

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

    3. Applied flip--62.4

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    7. Applied div-inv62.3

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

    8. Applied times-frac62.3

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

    9. Simplified62.5

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

    10. Taylor expanded around 0 2.2

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

    11. Simplified2.2

      \[\leadsto \color{blue}{\frac{-b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.9356897939138002 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))