Average Error: 33.8 → 6.7
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.95103770532986732 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.18338381295531773 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -7.70031330541463201 \cdot 10^{138}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.95103770532986732 \cdot 10^{-253}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 4.18338381295531773 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r177203 = b;
        double r177204 = -r177203;
        double r177205 = r177203 * r177203;
        double r177206 = 4.0;
        double r177207 = a;
        double r177208 = r177206 * r177207;
        double r177209 = c;
        double r177210 = r177208 * r177209;
        double r177211 = r177205 - r177210;
        double r177212 = sqrt(r177211);
        double r177213 = r177204 + r177212;
        double r177214 = 2.0;
        double r177215 = r177214 * r177207;
        double r177216 = r177213 / r177215;
        return r177216;
}
↓
double f(double a, double b, double c) {
        double r177217 = b;
        double r177218 = -7.700313305414632e+138;
        bool r177219 = r177217 <= r177218;
        double r177220 = 1.0;
        double r177221 = c;
        double r177222 = r177221 / r177217;
        double r177223 = a;
        double r177224 = r177217 / r177223;
        double r177225 = r177222 - r177224;
        double r177226 = r177220 * r177225;
        double r177227 = -3.9510377053298673e-253;
        bool r177228 = r177217 <= r177227;
        double r177229 = -r177217;
        double r177230 = r177217 * r177217;
        double r177231 = 4.0;
        double r177232 = r177231 * r177223;
        double r177233 = r177232 * r177221;
        double r177234 = r177230 - r177233;
        double r177235 = sqrt(r177234);
        double r177236 = r177229 + r177235;
        double r177237 = 2.0;
        double r177238 = r177237 * r177223;
        double r177239 = r177236 / r177238;
        double r177240 = 4.183383812955318e+98;
        bool r177241 = r177217 <= r177240;
        double r177242 = 1.0;
        double r177243 = r177237 * r177221;
        double r177244 = r177229 - r177235;
        double r177245 = r177243 / r177244;
        double r177246 = r177242 * r177245;
        double r177247 = -1.0;
        double r177248 = r177247 * r177222;
        double r177249 = r177241 ? r177246 : r177248;
        double r177250 = r177228 ? r177239 : r177249;
        double r177251 = r177219 ? r177226 : r177250;
        return r177251;
}

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

Original33.8
Target21.2
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.700313305414632e+138

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -7.700313305414632e+138 < b < -3.9510377053298673e-253

    1. Initial program 8.0

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

    if -3.9510377053298673e-253 < b < 4.183383812955318e+98

    1. Initial program 29.8

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

    3. Applied clear-num29.9

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

    5. Applied *-un-lft-identity29.9

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

    7. Applied flip-+29.9

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

    8. Applied associate-/r/30.0

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

    9. Applied associate-/r*30.0

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

    10. Simplified15.8

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

    11. Taylor expanded around 0 9.8

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

    if 4.183383812955318e+98 < b

    1. Initial program 59.5

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

    2. Taylor expanded around inf 2.8

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.95103770532986732 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.18338381295531773 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))