Average Error: 32.8 → 10.0
Time: 17.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -3.063397748446981 \cdot 10^{+71}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2275183 = b;
        double r2275184 = -r2275183;
        double r2275185 = r2275183 * r2275183;
        double r2275186 = 4.0;
        double r2275187 = a;
        double r2275188 = r2275186 * r2275187;
        double r2275189 = c;
        double r2275190 = r2275188 * r2275189;
        double r2275191 = r2275185 - r2275190;
        double r2275192 = sqrt(r2275191);
        double r2275193 = r2275184 + r2275192;
        double r2275194 = 2.0;
        double r2275195 = r2275194 * r2275187;
        double r2275196 = r2275193 / r2275195;
        return r2275196;
}
↓
double f(double a, double b, double c) {
        double r2275197 = b;
        double r2275198 = -3.063397748446981e+71;
        bool r2275199 = r2275197 <= r2275198;
        double r2275200 = c;
        double r2275201 = r2275200 / r2275197;
        double r2275202 = a;
        double r2275203 = r2275197 / r2275202;
        double r2275204 = r2275201 - r2275203;
        double r2275205 = 2.0;
        double r2275206 = r2275204 * r2275205;
        double r2275207 = r2275206 / r2275205;
        double r2275208 = 3.1295384133612364e-73;
        bool r2275209 = r2275197 <= r2275208;
        double r2275210 = 1.0;
        double r2275211 = r2275210 / r2275202;
        double r2275212 = r2275197 * r2275197;
        double r2275213 = 4.0;
        double r2275214 = r2275202 * r2275200;
        double r2275215 = r2275213 * r2275214;
        double r2275216 = r2275212 - r2275215;
        double r2275217 = sqrt(r2275216);
        double r2275218 = r2275217 - r2275197;
        double r2275219 = r2275211 * r2275218;
        double r2275220 = r2275219 / r2275205;
        double r2275221 = -2.0;
        double r2275222 = r2275221 * r2275201;
        double r2275223 = r2275222 / r2275205;
        double r2275224 = r2275209 ? r2275220 : r2275223;
        double r2275225 = r2275199 ? r2275207 : r2275224;
        return r2275225;
}

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

Original32.8
Target20.1
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.063397748446981e+71

    1. Initial program 38.6

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

    2. Simplified38.5

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

    3. Taylor expanded around -inf 4.7

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

    4. Simplified4.7

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

    if -3.063397748446981e+71 < b < 3.1295384133612364e-73

    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

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

    4. Applied div-inv13.2

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

    if 3.1295384133612364e-73 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    4. Applied div-inv52.3

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

    5. Taylor expanded around inf 9.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

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