Average Error: 33.4 → 9.9
Time: 18.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.148407540792454 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -5.148407540792454 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r747502 = b;
        double r747503 = -r747502;
        double r747504 = r747502 * r747502;
        double r747505 = 4.0;
        double r747506 = a;
        double r747507 = r747505 * r747506;
        double r747508 = c;
        double r747509 = r747507 * r747508;
        double r747510 = r747504 - r747509;
        double r747511 = sqrt(r747510);
        double r747512 = r747503 + r747511;
        double r747513 = 2.0;
        double r747514 = r747513 * r747506;
        double r747515 = r747512 / r747514;
        return r747515;
}
↓
double f(double a, double b, double c) {
        double r747516 = b;
        double r747517 = -5.148407540792454e+110;
        bool r747518 = r747516 <= r747517;
        double r747519 = c;
        double r747520 = r747519 / r747516;
        double r747521 = a;
        double r747522 = r747516 / r747521;
        double r747523 = r747520 - r747522;
        double r747524 = 2.0;
        double r747525 = r747523 * r747524;
        double r747526 = r747525 / r747524;
        double r747527 = 2.326372645943808e-74;
        bool r747528 = r747516 <= r747527;
        double r747529 = 1.0;
        double r747530 = r747529 / r747521;
        double r747531 = r747516 * r747516;
        double r747532 = 4.0;
        double r747533 = r747519 * r747521;
        double r747534 = r747532 * r747533;
        double r747535 = r747531 - r747534;
        double r747536 = sqrt(r747535);
        double r747537 = r747530 * r747536;
        double r747538 = r747537 - r747522;
        double r747539 = r747538 / r747524;
        double r747540 = -2.0;
        double r747541 = r747520 * r747540;
        double r747542 = r747541 / r747524;
        double r747543 = r747528 ? r747539 : r747542;
        double r747544 = r747518 ? r747526 : r747543;
        return r747544;
}

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 < -5.148407540792454e+110

    1. Initial program 46.9

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

    2. Simplified46.9

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

    4. Applied div-sub46.9

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

    6. Applied div-inv47.0

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

    7. Taylor expanded around -inf 3.6

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

    8. Simplified3.6

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

    if -5.148407540792454e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.7

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

    4. Applied div-sub12.7

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

    6. Applied div-inv12.8

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    4. Applied div-sub53.2

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

    6. Applied div-inv54.1

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

    7. Taylor expanded around inf 8.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.148407540792454 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))