Average Error: 33.0 → 10.8
Time: 26.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{1}{\frac{2}{-2 \cdot \frac{c}{b}}}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 r2000532 = b;
        double r2000533 = -r2000532;
        double r2000534 = r2000532 * r2000532;
        double r2000535 = 4.0;
        double r2000536 = a;
        double r2000537 = r2000535 * r2000536;
        double r2000538 = c;
        double r2000539 = r2000537 * r2000538;
        double r2000540 = r2000534 - r2000539;
        double r2000541 = sqrt(r2000540);
        double r2000542 = r2000533 + r2000541;
        double r2000543 = 2.0;
        double r2000544 = r2000543 * r2000536;
        double r2000545 = r2000542 / r2000544;
        return r2000545;
}
↓
double f(double a, double b, double c) {
        double r2000546 = b;
        double r2000547 = -9.348931433494438e+39;
        bool r2000548 = r2000546 <= r2000547;
        double r2000549 = c;
        double r2000550 = r2000549 / r2000546;
        double r2000551 = a;
        double r2000552 = r2000546 / r2000551;
        double r2000553 = r2000550 - r2000552;
        double r2000554 = 1.3353078790738604e-121;
        bool r2000555 = r2000546 <= r2000554;
        double r2000556 = -4.0;
        double r2000557 = r2000551 * r2000556;
        double r2000558 = r2000557 * r2000549;
        double r2000559 = r2000546 * r2000546;
        double r2000560 = r2000558 + r2000559;
        double r2000561 = sqrt(r2000560);
        double r2000562 = r2000561 - r2000546;
        double r2000563 = 2.0;
        double r2000564 = r2000562 / r2000563;
        double r2000565 = r2000564 / r2000551;
        double r2000566 = 1.6168702840263923e-79;
        bool r2000567 = r2000546 <= r2000566;
        double r2000568 = 1.0;
        double r2000569 = -2.0;
        double r2000570 = r2000569 * r2000550;
        double r2000571 = r2000563 / r2000570;
        double r2000572 = r2000568 / r2000571;
        double r2000573 = 1.546013236023957e-67;
        bool r2000574 = r2000546 <= r2000573;
        double r2000575 = -r2000550;
        double r2000576 = r2000574 ? r2000565 : r2000575;
        double r2000577 = r2000567 ? r2000572 : r2000576;
        double r2000578 = r2000555 ? r2000565 : r2000577;
        double r2000579 = r2000548 ? r2000553 : r2000578;
        return r2000579;
}

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 < -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 "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))