Average Error: 33.0 → 10.8
Time: 21.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{\frac{2}{-2 \cdot \frac{c}{b}}}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\

\mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{1}{\frac{2}{-2 \cdot \frac{c}{b}}}\\

\mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r13047362 = b;
        double r13047363 = -r13047362;
        double r13047364 = r13047362 * r13047362;
        double r13047365 = 4.0;
        double r13047366 = a;
        double r13047367 = r13047365 * r13047366;
        double r13047368 = c;
        double r13047369 = r13047367 * r13047368;
        double r13047370 = r13047364 - r13047369;
        double r13047371 = sqrt(r13047370);
        double r13047372 = r13047363 + r13047371;
        double r13047373 = 2.0;
        double r13047374 = r13047373 * r13047366;
        double r13047375 = r13047372 / r13047374;
        return r13047375;
}
↓
double f(double a, double b, double c) {
        double r13047376 = b;
        double r13047377 = -9.348931433494438e+39;
        bool r13047378 = r13047376 <= r13047377;
        double r13047379 = c;
        double r13047380 = r13047379 / r13047376;
        double r13047381 = a;
        double r13047382 = r13047376 / r13047381;
        double r13047383 = r13047380 - r13047382;
        double r13047384 = 1.3353078790738604e-121;
        bool r13047385 = r13047376 <= r13047384;
        double r13047386 = -4.0;
        double r13047387 = r13047381 * r13047386;
        double r13047388 = r13047387 * r13047379;
        double r13047389 = r13047376 * r13047376;
        double r13047390 = r13047388 + r13047389;
        double r13047391 = sqrt(r13047390);
        double r13047392 = r13047391 - r13047376;
        double r13047393 = 2.0;
        double r13047394 = r13047392 / r13047393;
        double r13047395 = r13047394 / r13047381;
        double r13047396 = 1.6168702840263923e-79;
        bool r13047397 = r13047376 <= r13047396;
        double r13047398 = 1.0;
        double r13047399 = -2.0;
        double r13047400 = r13047399 * r13047380;
        double r13047401 = r13047393 / r13047400;
        double r13047402 = r13047398 / r13047401;
        double r13047403 = 1.546013236023957e-67;
        bool r13047404 = r13047376 <= r13047403;
        double r13047405 = -r13047380;
        double r13047406 = r13047404 ? r13047395 : r13047405;
        double r13047407 = r13047397 ? r13047402 : r13047406;
        double r13047408 = r13047385 ? r13047395 : r13047407;
        double r13047409 = r13047378 ? r13047383 : r13047408;
        return r13047409;
}

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

Original33.0
Target20.1
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Taylor expanded around -inf 6.2

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

    if -9.348931433494438e+39 < b < 1.3353078790738604e-121 or 1.6168702840263923e-79 < b < 1.546013236023957e-67

    1. Initial program 12.9

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

    3. Applied div-inv13.0

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

    4. Simplified13.0

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

    6. Applied associate-*r/12.9

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

    7. Simplified12.9

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

    if 1.3353078790738604e-121 < b < 1.6168702840263923e-79

    1. Initial program 32.1

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

    3. Applied clear-num32.1

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

    4. Simplified32.1

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

    5. Taylor expanded around inf 35.8

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{-2 \cdot \frac{c}{b}}}}\]

    if 1.546013236023957e-67 < b

    1. Initial program 52.3

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

    2. Taylor expanded around inf 9.2

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

    3. Simplified9.2

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{\frac{2}{-2 \cdot \frac{c}{b}}}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))