Average Error: 34.2 → 6.7
Time: 14.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.519893370196745925426926386855899506479 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.684322567234166107344691100808795296642 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{4}{1} \cdot c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.519893370196745925426926386855899506479 \cdot 10^{-307}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 9.684322567234166107344691100808795296642 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{\frac{2}{\frac{4}{1} \cdot c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79204 = b;
        double r79205 = -r79204;
        double r79206 = r79204 * r79204;
        double r79207 = 4.0;
        double r79208 = a;
        double r79209 = c;
        double r79210 = r79208 * r79209;
        double r79211 = r79207 * r79210;
        double r79212 = r79206 - r79211;
        double r79213 = sqrt(r79212);
        double r79214 = r79205 + r79213;
        double r79215 = 2.0;
        double r79216 = r79215 * r79208;
        double r79217 = r79214 / r79216;
        return r79217;
}
↓
double f(double a, double b, double c) {
        double r79218 = b;
        double r79219 = -1.8440038131758226e+119;
        bool r79220 = r79218 <= r79219;
        double r79221 = 1.0;
        double r79222 = c;
        double r79223 = r79222 / r79218;
        double r79224 = a;
        double r79225 = r79218 / r79224;
        double r79226 = r79223 - r79225;
        double r79227 = r79221 * r79226;
        double r79228 = -1.519893370196746e-307;
        bool r79229 = r79218 <= r79228;
        double r79230 = -r79218;
        double r79231 = r79218 * r79218;
        double r79232 = 4.0;
        double r79233 = r79224 * r79222;
        double r79234 = r79232 * r79233;
        double r79235 = r79231 - r79234;
        double r79236 = sqrt(r79235);
        double r79237 = r79230 + r79236;
        double r79238 = 1.0;
        double r79239 = 2.0;
        double r79240 = r79239 * r79224;
        double r79241 = r79238 / r79240;
        double r79242 = r79237 * r79241;
        double r79243 = 9.684322567234166e+98;
        bool r79244 = r79218 <= r79243;
        double r79245 = r79232 / r79238;
        double r79246 = r79245 * r79222;
        double r79247 = r79239 / r79246;
        double r79248 = r79230 - r79236;
        double r79249 = r79247 * r79248;
        double r79250 = r79238 / r79249;
        double r79251 = -1.0;
        double r79252 = r79251 * r79223;
        double r79253 = r79244 ? r79250 : r79252;
        double r79254 = r79229 ? r79242 : r79253;
        double r79255 = r79220 ? r79227 : r79254;
        return r79255;
}

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

Original34.2
Target21.1
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8440038131758226e+119

    1. Initial program 51.6

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

    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 -1.8440038131758226e+119 < b < -1.519893370196746e-307

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if -1.519893370196746e-307 < b < 9.684322567234166e+98

    1. Initial program 32.2

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

    3. Applied flip-+32.2

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

    4. Simplified16.7

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

    6. Applied clear-num16.9

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

    7. Simplified16.2

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

    9. Applied associate-/l*16.3

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

    10. Simplified9.6

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

    if 9.684322567234166e+98 < b

    1. Initial program 59.3

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

    2. Taylor expanded around inf 2.6

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.519893370196745925426926386855899506479 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.684322567234166107344691100808795296642 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{4}{1} \cdot c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))