Average Error: 34.2 → 9.1
Time: 18.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -763129212434271441067123993682640896:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\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 \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r79211 = b;
        double r79212 = -r79211;
        double r79213 = r79211 * r79211;
        double r79214 = 4.0;
        double r79215 = a;
        double r79216 = c;
        double r79217 = r79215 * r79216;
        double r79218 = r79214 * r79217;
        double r79219 = r79213 - r79218;
        double r79220 = sqrt(r79219);
        double r79221 = r79212 - r79220;
        double r79222 = 2.0;
        double r79223 = r79222 * r79215;
        double r79224 = r79221 / r79223;
        return r79224;
}
↓
double f(double a, double b, double c) {
        double r79225 = b;
        double r79226 = -7.631292124342714e+35;
        bool r79227 = r79225 <= r79226;
        double r79228 = -1.0;
        double r79229 = c;
        double r79230 = r79229 / r79225;
        double r79231 = r79228 * r79230;
        double r79232 = 9.580019013081131e-278;
        bool r79233 = r79225 <= r79232;
        double r79234 = 4.0;
        double r79235 = r79234 * r79229;
        double r79236 = a;
        double r79237 = r79235 * r79236;
        double r79238 = r79225 * r79225;
        double r79239 = r79236 * r79229;
        double r79240 = r79234 * r79239;
        double r79241 = r79238 - r79240;
        double r79242 = sqrt(r79241);
        double r79243 = r79242 - r79225;
        double r79244 = r79237 / r79243;
        double r79245 = 2.0;
        double r79246 = r79245 * r79236;
        double r79247 = r79244 / r79246;
        double r79248 = 5.031608061939103e+53;
        bool r79249 = r79225 <= r79248;
        double r79250 = -r79225;
        double r79251 = r79250 - r79242;
        double r79252 = 1.0;
        double r79253 = r79252 / r79246;
        double r79254 = r79251 * r79253;
        double r79255 = 1.0;
        double r79256 = r79225 / r79236;
        double r79257 = r79230 - r79256;
        double r79258 = r79255 * r79257;
        double r79259 = r79249 ? r79254 : r79258;
        double r79260 = r79233 ? r79247 : r79259;
        double r79261 = r79227 ? r79231 : r79260;
        return r79261;
}

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
Target21.3
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.631292124342714e+35

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 4.5

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

    if -7.631292124342714e+35 < b < 9.580019013081131e-278

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified16.7

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

    5. Simplified16.7

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

    if 9.580019013081131e-278 < b < 5.031608061939103e+53

    1. Initial program 9.4

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

    3. Applied div-inv9.6

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

    if 5.031608061939103e+53 < b

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

    3. Simplified5.7

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))