Average Error: 34.2 → 6.7
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r57613 = b;
        double r57614 = -r57613;
        double r57615 = r57613 * r57613;
        double r57616 = 4.0;
        double r57617 = a;
        double r57618 = r57616 * r57617;
        double r57619 = c;
        double r57620 = r57618 * r57619;
        double r57621 = r57615 - r57620;
        double r57622 = sqrt(r57621);
        double r57623 = r57614 + r57622;
        double r57624 = 2.0;
        double r57625 = r57624 * r57617;
        double r57626 = r57623 / r57625;
        return r57626;
}
↓
double f(double a, double b, double c) {
        double r57627 = b;
        double r57628 = -4.739386840053889e+131;
        bool r57629 = r57627 <= r57628;
        double r57630 = 1.0;
        double r57631 = c;
        double r57632 = r57631 / r57627;
        double r57633 = a;
        double r57634 = r57627 / r57633;
        double r57635 = r57632 - r57634;
        double r57636 = r57630 * r57635;
        double r57637 = -2.1023086245622604e-293;
        bool r57638 = r57627 <= r57637;
        double r57639 = -r57627;
        double r57640 = r57627 * r57627;
        double r57641 = 4.0;
        double r57642 = r57641 * r57633;
        double r57643 = r57642 * r57631;
        double r57644 = r57640 - r57643;
        double r57645 = sqrt(r57644);
        double r57646 = r57639 + r57645;
        double r57647 = 1.0;
        double r57648 = 2.0;
        double r57649 = r57648 * r57633;
        double r57650 = r57647 / r57649;
        double r57651 = r57646 * r57650;
        double r57652 = 6.09240124692818e+90;
        bool r57653 = r57627 <= r57652;
        double r57654 = 0.5;
        double r57655 = r57654 / r57631;
        double r57656 = r57639 - r57645;
        double r57657 = r57655 * r57656;
        double r57658 = r57647 / r57657;
        double r57659 = -1.0;
        double r57660 = r57659 * r57632;
        double r57661 = r57653 ? r57658 : r57660;
        double r57662 = r57638 ? r57651 : r57661;
        double r57663 = r57629 ? r57636 : r57662;
        return r57663;
}

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 4 regimes

  2. if b < -4.739386840053889e+131

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 2.4

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

    3. Simplified2.4

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

    if -4.739386840053889e+131 < b < -2.1023086245622604e-293

    1. Initial program 9.2

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

    3. Applied div-inv9.4

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

    if -2.1023086245622604e-293 < b < 6.09240124692818e+90

    1. Initial program 31.3

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

    3. Applied flip-+31.3

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

    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. Simplified15.6

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

    8. Taylor expanded around 0 8.8

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

    if 6.09240124692818e+90 < b

    1. Initial program 59.2

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))