Average Error: 34.1 → 10.1
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.46337219442650475190360405115215330559 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.26844514409972828090140298620599613013 \cdot 10^{-106}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.46337219442650475190360405115215330559 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.26844514409972828090140298620599613013 \cdot 10^{-106}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r57145 = b;
        double r57146 = -r57145;
        double r57147 = r57145 * r57145;
        double r57148 = 4.0;
        double r57149 = a;
        double r57150 = c;
        double r57151 = r57149 * r57150;
        double r57152 = r57148 * r57151;
        double r57153 = r57147 - r57152;
        double r57154 = sqrt(r57153);
        double r57155 = r57146 + r57154;
        double r57156 = 2.0;
        double r57157 = r57156 * r57149;
        double r57158 = r57155 / r57157;
        return r57158;
}
↓
double f(double a, double b, double c) {
        double r57159 = b;
        double r57160 = -2.463372194426505e+111;
        bool r57161 = r57159 <= r57160;
        double r57162 = 1.0;
        double r57163 = c;
        double r57164 = r57163 / r57159;
        double r57165 = a;
        double r57166 = r57159 / r57165;
        double r57167 = r57164 - r57166;
        double r57168 = r57162 * r57167;
        double r57169 = 6.268445144099728e-106;
        bool r57170 = r57159 <= r57169;
        double r57171 = -r57159;
        double r57172 = r57159 * r57159;
        double r57173 = 4.0;
        double r57174 = r57165 * r57163;
        double r57175 = r57173 * r57174;
        double r57176 = r57172 - r57175;
        double r57177 = sqrt(r57176);
        double r57178 = r57171 + r57177;
        double r57179 = 1.0;
        double r57180 = 2.0;
        double r57181 = r57180 * r57165;
        double r57182 = r57179 / r57181;
        double r57183 = r57178 * r57182;
        double r57184 = -1.0;
        double r57185 = r57184 * r57164;
        double r57186 = r57170 ? r57183 : r57185;
        double r57187 = r57161 ? r57168 : r57186;
        return r57187;
}

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

Original34.1
Target21.0
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.463372194426505e+111

    1. Initial program 48.7

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

    2. Taylor expanded around -inf 3.0

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

    3. Simplified3.0

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

    if -2.463372194426505e+111 < b < 6.268445144099728e-106

    1. Initial program 11.5

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

    3. Applied div-inv11.6

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

    if 6.268445144099728e-106 < b

    1. Initial program 52.4

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

    2. Taylor expanded around inf 10.9

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.46337219442650475190360405115215330559 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.26844514409972828090140298620599613013 \cdot 10^{-106}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))