Average Error: 33.6 → 9.7
Time: 27.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -4.694684309811035 \cdot 10^{+121}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3849163 = b;
        double r3849164 = -r3849163;
        double r3849165 = r3849163 * r3849163;
        double r3849166 = 4.0;
        double r3849167 = a;
        double r3849168 = r3849166 * r3849167;
        double r3849169 = c;
        double r3849170 = r3849168 * r3849169;
        double r3849171 = r3849165 - r3849170;
        double r3849172 = sqrt(r3849171);
        double r3849173 = r3849164 + r3849172;
        double r3849174 = 2.0;
        double r3849175 = r3849174 * r3849167;
        double r3849176 = r3849173 / r3849175;
        return r3849176;
}
↓
double f(double a, double b, double c) {
        double r3849177 = b;
        double r3849178 = -4.694684309811035e+121;
        bool r3849179 = r3849177 <= r3849178;
        double r3849180 = c;
        double r3849181 = r3849180 / r3849177;
        double r3849182 = a;
        double r3849183 = r3849177 / r3849182;
        double r3849184 = r3849181 - r3849183;
        double r3849185 = 2.0;
        double r3849186 = r3849184 * r3849185;
        double r3849187 = r3849186 / r3849185;
        double r3849188 = 4.6659701943749105e-84;
        bool r3849189 = r3849177 <= r3849188;
        double r3849190 = 1.0;
        double r3849191 = r3849190 / r3849182;
        double r3849192 = r3849177 * r3849177;
        double r3849193 = 4.0;
        double r3849194 = r3849180 * r3849182;
        double r3849195 = r3849193 * r3849194;
        double r3849196 = r3849192 - r3849195;
        double r3849197 = sqrt(r3849196);
        double r3849198 = r3849197 - r3849177;
        double r3849199 = r3849191 * r3849198;
        double r3849200 = r3849199 / r3849185;
        double r3849201 = -2.0;
        double r3849202 = r3849181 * r3849201;
        double r3849203 = r3849202 / r3849185;
        double r3849204 = r3849189 ? r3849200 : r3849203;
        double r3849205 = r3849179 ? r3849187 : r3849204;
        return r3849205;
}

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
Target20.4
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.694684309811035e+121

    1. Initial program 49.8

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

    2. Simplified49.8

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

    4. Applied div-inv49.9

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

    5. Taylor expanded around -inf 2.6

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

    6. Simplified2.6

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

    if -4.694684309811035e+121 < b < 4.6659701943749105e-84

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied div-inv12.3

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

    if 4.6659701943749105e-84 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))