Average Error: 34.3 → 7.2
Time: 5.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.4397859828859872 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.0621980184587312 \cdot 10^{-219}:\\ \;\;\;\;\frac{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.66563346804975556 \cdot 10^{146}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot 4\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.4397859828859872 \cdot 10^{117}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.0621980184587312 \cdot 10^{-219}:\\
\;\;\;\;\frac{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.66563346804975556 \cdot 10^{146}:\\
\;\;\;\;\frac{\left(\frac{1}{2} \cdot 4\right) \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 r141223 = b;
        double r141224 = -r141223;
        double r141225 = r141223 * r141223;
        double r141226 = 4.0;
        double r141227 = a;
        double r141228 = r141226 * r141227;
        double r141229 = c;
        double r141230 = r141228 * r141229;
        double r141231 = r141225 - r141230;
        double r141232 = sqrt(r141231);
        double r141233 = r141224 + r141232;
        double r141234 = 2.0;
        double r141235 = r141234 * r141227;
        double r141236 = r141233 / r141235;
        return r141236;
}
↓
double f(double a, double b, double c) {
        double r141237 = b;
        double r141238 = -3.439785982885987e+117;
        bool r141239 = r141237 <= r141238;
        double r141240 = 1.0;
        double r141241 = c;
        double r141242 = r141241 / r141237;
        double r141243 = a;
        double r141244 = r141237 / r141243;
        double r141245 = r141242 - r141244;
        double r141246 = r141240 * r141245;
        double r141247 = -1.0621980184587312e-219;
        bool r141248 = r141237 <= r141247;
        double r141249 = -r141237;
        double r141250 = r141237 * r141237;
        double r141251 = 4.0;
        double r141252 = r141251 * r141243;
        double r141253 = r141252 * r141241;
        double r141254 = r141250 - r141253;
        double r141255 = sqrt(r141254);
        double r141256 = r141249 + r141255;
        double r141257 = log(r141256);
        double r141258 = exp(r141257);
        double r141259 = 2.0;
        double r141260 = r141259 * r141243;
        double r141261 = r141258 / r141260;
        double r141262 = 1.6656334680497556e+146;
        bool r141263 = r141237 <= r141262;
        double r141264 = 1.0;
        double r141265 = r141264 / r141259;
        double r141266 = r141265 * r141251;
        double r141267 = r141266 * r141241;
        double r141268 = r141249 - r141255;
        double r141269 = r141267 / r141268;
        double r141270 = -1.0;
        double r141271 = r141270 * r141242;
        double r141272 = r141263 ? r141269 : r141271;
        double r141273 = r141248 ? r141261 : r141272;
        double r141274 = r141239 ? r141246 : r141273;
        return r141274;
}

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

Original34.3
Target21.1
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.439785982885987e+117

    1. Initial program 51.7

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

    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 -3.439785982885987e+117 < b < -1.0621980184587312e-219

    1. Initial program 6.7

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

    3. Applied add-exp-log10.5

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

    if -1.0621980184587312e-219 < b < 1.6656334680497556e+146

    1. Initial program 32.0

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

    3. Applied flip-+32.0

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

    4. Simplified16.0

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

    6. Applied clear-num16.2

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

    7. Simplified15.0

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

    9. Applied associate-/l*15.1

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

    10. Simplified9.6

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

    12. Applied associate-/r*9.1

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

    13. Simplified9.0

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

    if 1.6656334680497556e+146 < b

    1. Initial program 63.3

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.4397859828859872 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.0621980184587312 \cdot 10^{-219}:\\ \;\;\;\;\frac{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.66563346804975556 \cdot 10^{146}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot 4\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))