Average Error: 19.7 → 7.8
Time: 5.4s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.031354366652439074047233382039797740753 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.947510423350864326592355394008700948277 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \le 6.614815782594423881712543712874288384103 \cdot 10^{53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.031354366652439074047233382039797740753 \cdot 10^{153}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 2.947510423350864326592355394008700948277 \cdot 10^{-310}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}\\

\mathbf{elif}\;b \le 6.614815782594423881712543712874288384103 \cdot 10^{53}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\end{array}
double f(double a, double b, double c) {
        double r34462 = b;
        double r34463 = 0.0;
        bool r34464 = r34462 >= r34463;
        double r34465 = -r34462;
        double r34466 = r34462 * r34462;
        double r34467 = 4.0;
        double r34468 = a;
        double r34469 = r34467 * r34468;
        double r34470 = c;
        double r34471 = r34469 * r34470;
        double r34472 = r34466 - r34471;
        double r34473 = sqrt(r34472);
        double r34474 = r34465 - r34473;
        double r34475 = 2.0;
        double r34476 = r34475 * r34468;
        double r34477 = r34474 / r34476;
        double r34478 = r34475 * r34470;
        double r34479 = r34465 + r34473;
        double r34480 = r34478 / r34479;
        double r34481 = r34464 ? r34477 : r34480;
        return r34481;
}
↓
double f(double a, double b, double c) {
        double r34482 = b;
        double r34483 = -7.031354366652439e+153;
        bool r34484 = r34482 <= r34483;
        double r34485 = 0.0;
        bool r34486 = r34482 >= r34485;
        double r34487 = -r34482;
        double r34488 = r34482 * r34482;
        double r34489 = 4.0;
        double r34490 = a;
        double r34491 = r34489 * r34490;
        double r34492 = c;
        double r34493 = r34491 * r34492;
        double r34494 = r34488 - r34493;
        double r34495 = sqrt(r34494);
        double r34496 = r34487 - r34495;
        double r34497 = 2.0;
        double r34498 = r34497 * r34490;
        double r34499 = r34496 / r34498;
        double r34500 = r34497 * r34492;
        double r34501 = r34490 * r34492;
        double r34502 = r34501 / r34482;
        double r34503 = r34497 * r34502;
        double r34504 = r34503 - r34482;
        double r34505 = r34487 + r34504;
        double r34506 = r34500 / r34505;
        double r34507 = r34486 ? r34499 : r34506;
        double r34508 = 2.94751042335086e-310;
        bool r34509 = r34482 <= r34508;
        double r34510 = 1.0;
        double r34511 = r34482 - r34503;
        double r34512 = r34487 - r34511;
        double r34513 = r34498 / r34512;
        double r34514 = r34510 / r34513;
        double r34515 = r34487 + r34495;
        double r34516 = r34500 / r34515;
        double r34517 = r34486 ? r34514 : r34516;
        double r34518 = 6.614815782594424e+53;
        bool r34519 = r34482 <= r34518;
        double r34520 = 1.0;
        double r34521 = r34492 / r34482;
        double r34522 = r34482 / r34490;
        double r34523 = r34521 - r34522;
        double r34524 = r34520 * r34523;
        double r34525 = 0.0;
        double r34526 = r34489 * r34501;
        double r34527 = r34525 + r34526;
        double r34528 = r34527 / r34496;
        double r34529 = r34500 / r34528;
        double r34530 = r34486 ? r34524 : r34529;
        double r34531 = r34519 ? r34507 : r34530;
        double r34532 = r34509 ? r34517 : r34531;
        double r34533 = r34484 ? r34507 : r34532;
        return r34533;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -7.031354366652439e+153 or 2.94751042335086e-310 < b < 6.614815782594424e+53

    1. Initial program 21.2

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

    2. Taylor expanded around -inf 8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]

    if -7.031354366652439e+153 < b < 2.94751042335086e-310

    1. Initial program 8.2

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

    2. Taylor expanded around inf 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    3. Using strategy rm

    4. Applied clear-num8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    if 6.614815782594424e+53 < b

    1. Initial program 37.9

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

    2. Taylor expanded around inf 10.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Taylor expanded around 0 5.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    4. Simplified5.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied flip-+5.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    7. Simplified5.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.031354366652439074047233382039797740753 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.947510423350864326592355394008700948277 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \le 6.614815782594423881712543712874288384103 \cdot 10^{53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))