Average Error: 34.3 → 13.2
Time: 18.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.304421310335197068961304849785779948437 \cdot 10^{-75}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.680510304999259194268524546555599685222 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.304421310335197068961304849785779948437 \cdot 10^{-75}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.680510304999259194268524546555599685222 \cdot 10^{-106}:\\
\;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r3675256 = b;
        double r3675257 = -r3675256;
        double r3675258 = r3675256 * r3675256;
        double r3675259 = 4.0;
        double r3675260 = a;
        double r3675261 = c;
        double r3675262 = r3675260 * r3675261;
        double r3675263 = r3675259 * r3675262;
        double r3675264 = r3675258 - r3675263;
        double r3675265 = sqrt(r3675264);
        double r3675266 = r3675257 - r3675265;
        double r3675267 = 2.0;
        double r3675268 = r3675267 * r3675260;
        double r3675269 = r3675266 / r3675268;
        return r3675269;
}
↓
double f(double a, double b, double c) {
        double r3675270 = b;
        double r3675271 = -3.304421310335197e-75;
        bool r3675272 = r3675270 <= r3675271;
        double r3675273 = -1.0;
        double r3675274 = c;
        double r3675275 = r3675274 / r3675270;
        double r3675276 = r3675273 * r3675275;
        double r3675277 = 1.6805103049992592e-106;
        bool r3675278 = r3675270 <= r3675277;
        double r3675279 = a;
        double r3675280 = r3675274 * r3675279;
        double r3675281 = 4.0;
        double r3675282 = r3675280 * r3675281;
        double r3675283 = -r3675270;
        double r3675284 = r3675270 * r3675270;
        double r3675285 = r3675284 - r3675282;
        double r3675286 = sqrt(r3675285);
        double r3675287 = r3675283 + r3675286;
        double r3675288 = r3675282 / r3675287;
        double r3675289 = 2.0;
        double r3675290 = r3675279 * r3675289;
        double r3675291 = r3675288 / r3675290;
        double r3675292 = 1.0;
        double r3675293 = r3675270 / r3675279;
        double r3675294 = r3675275 - r3675293;
        double r3675295 = r3675292 * r3675294;
        double r3675296 = r3675278 ? r3675291 : r3675295;
        double r3675297 = r3675272 ? r3675276 : r3675296;
        return r3675297;
}

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
Target21.3
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.304421310335197e-75

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 9.8

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

    if -3.304421310335197e-75 < b < 1.6805103049992592e-106

    1. Initial program 17.6

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

    3. Applied flip--20.2

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

    4. Taylor expanded around inf 19.1

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

    if 1.6805103049992592e-106 < b

    1. Initial program 25.3

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

    2. Taylor expanded around inf 12.5

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

    3. Simplified12.5

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.304421310335197068961304849785779948437 \cdot 10^{-75}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.680510304999259194268524546555599685222 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot 4}{\left(-b\right) + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))