Average Error: 33.9 → 8.8
Time: 7.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.896807826547109987464579044865965135876 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 1.505965056562671542641677182626911662098 \cdot 10^{74}:\\ \;\;\;\;\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.896807826547109987464579044865965135876 \cdot 10^{-222}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 1.505965056562671542641677182626911662098 \cdot 10^{74}:\\
\;\;\;\;\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r86131 = b;
        double r86132 = -r86131;
        double r86133 = r86131 * r86131;
        double r86134 = 4.0;
        double r86135 = a;
        double r86136 = r86134 * r86135;
        double r86137 = c;
        double r86138 = r86136 * r86137;
        double r86139 = r86133 - r86138;
        double r86140 = sqrt(r86139);
        double r86141 = r86132 + r86140;
        double r86142 = 2.0;
        double r86143 = r86142 * r86135;
        double r86144 = r86141 / r86143;
        return r86144;
}
↓
double f(double a, double b, double c) {
        double r86145 = b;
        double r86146 = -1.361733299857302e+105;
        bool r86147 = r86145 <= r86146;
        double r86148 = 1.0;
        double r86149 = c;
        double r86150 = r86149 / r86145;
        double r86151 = a;
        double r86152 = r86145 / r86151;
        double r86153 = r86150 - r86152;
        double r86154 = r86148 * r86153;
        double r86155 = 6.89680782654711e-222;
        bool r86156 = r86145 <= r86155;
        double r86157 = 1.0;
        double r86158 = 2.0;
        double r86159 = r86158 * r86151;
        double r86160 = -r86145;
        double r86161 = r86145 * r86145;
        double r86162 = 4.0;
        double r86163 = r86162 * r86151;
        double r86164 = r86163 * r86149;
        double r86165 = r86161 - r86164;
        double r86166 = sqrt(r86165);
        double r86167 = r86160 + r86166;
        double r86168 = r86159 / r86167;
        double r86169 = r86157 / r86168;
        double r86170 = 1.5059650565626715e+74;
        bool r86171 = r86145 <= r86170;
        double r86172 = 0.0;
        double r86173 = r86172 + r86164;
        double r86174 = r86160 - r86166;
        double r86175 = r86173 / r86174;
        double r86176 = r86157 / r86159;
        double r86177 = r86175 * r86176;
        double r86178 = -1.0;
        double r86179 = r86178 * r86150;
        double r86180 = r86171 ? r86177 : r86179;
        double r86181 = r86156 ? r86169 : r86180;
        double r86182 = r86147 ? r86154 : r86181;
        return r86182;
}

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

Original33.9
Target21.1
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.361733299857302e+105 < b < 6.89680782654711e-222

    1. Initial program 9.6

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

    3. Applied clear-num9.8

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

    if 6.89680782654711e-222 < b < 1.5059650565626715e+74

    1. Initial program 33.9

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

    3. Applied flip-+34.0

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

    4. Simplified16.5

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv16.5

      \[\leadsto \color{blue}{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}}\]

    if 1.5059650565626715e+74 < b

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.896807826547109987464579044865965135876 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 1.505965056562671542641677182626911662098 \cdot 10^{74}:\\ \;\;\;\;\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))