Average Error: 33.7 → 10.9
Time: 9.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r91164 = b;
        double r91165 = -r91164;
        double r91166 = r91164 * r91164;
        double r91167 = 4.0;
        double r91168 = a;
        double r91169 = c;
        double r91170 = r91168 * r91169;
        double r91171 = r91167 * r91170;
        double r91172 = r91166 - r91171;
        double r91173 = sqrt(r91172);
        double r91174 = r91165 + r91173;
        double r91175 = 2.0;
        double r91176 = r91175 * r91168;
        double r91177 = r91174 / r91176;
        return r91177;
}
↓
double f(double a, double b, double c) {
        double r91178 = b;
        double r91179 = -1.9827654008890006e+134;
        bool r91180 = r91178 <= r91179;
        double r91181 = 1.0;
        double r91182 = c;
        double r91183 = r91182 / r91178;
        double r91184 = a;
        double r91185 = r91178 / r91184;
        double r91186 = r91183 - r91185;
        double r91187 = r91181 * r91186;
        double r91188 = 1.1860189201379418e-161;
        bool r91189 = r91178 <= r91188;
        double r91190 = r91178 * r91178;
        double r91191 = 4.0;
        double r91192 = r91184 * r91182;
        double r91193 = r91191 * r91192;
        double r91194 = r91190 - r91193;
        double r91195 = sqrt(r91194);
        double r91196 = r91195 - r91178;
        double r91197 = 1.0;
        double r91198 = 2.0;
        double r91199 = r91197 / r91198;
        double r91200 = r91199 / r91184;
        double r91201 = r91196 * r91200;
        double r91202 = -1.0;
        double r91203 = r91202 * r91183;
        double r91204 = r91189 ? r91201 : r91203;
        double r91205 = r91180 ? r91187 : r91204;
        return r91205;
}

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

Original33.7
Target21.0
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Simplified56.8

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

    3. Taylor expanded around -inf 3.1

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

    4. Simplified3.1

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

    if -1.9827654008890006e+134 < b < 1.1860189201379418e-161

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied *-un-lft-identity10.3

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

    5. Applied div-inv10.3

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

    6. Applied times-frac10.5

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

    7. Simplified10.5

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

    if 1.1860189201379418e-161 < b

    1. Initial program 49.6

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

    2. Simplified49.6

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

    3. Taylor expanded around inf 13.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))