Average Error: 34.4 → 10.7
Time: 20.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1495430 = b;
        double r1495431 = -r1495430;
        double r1495432 = r1495430 * r1495430;
        double r1495433 = 4.0;
        double r1495434 = a;
        double r1495435 = r1495433 * r1495434;
        double r1495436 = c;
        double r1495437 = r1495435 * r1495436;
        double r1495438 = r1495432 - r1495437;
        double r1495439 = sqrt(r1495438);
        double r1495440 = r1495431 + r1495439;
        double r1495441 = 2.0;
        double r1495442 = r1495441 * r1495434;
        double r1495443 = r1495440 / r1495442;
        return r1495443;
}
↓
double f(double a, double b, double c) {
        double r1495444 = b;
        double r1495445 = -5.6179139475653e+116;
        bool r1495446 = r1495444 <= r1495445;
        double r1495447 = c;
        double r1495448 = r1495447 / r1495444;
        double r1495449 = a;
        double r1495450 = r1495444 / r1495449;
        double r1495451 = r1495448 - r1495450;
        double r1495452 = 1.0;
        double r1495453 = r1495451 * r1495452;
        double r1495454 = 2.8983489306952693e-35;
        bool r1495455 = r1495444 <= r1495454;
        double r1495456 = r1495444 * r1495444;
        double r1495457 = 4.0;
        double r1495458 = r1495449 * r1495447;
        double r1495459 = r1495457 * r1495458;
        double r1495460 = r1495456 - r1495459;
        double r1495461 = sqrt(r1495460);
        double r1495462 = sqrt(r1495461);
        double r1495463 = r1495462 * r1495462;
        double r1495464 = r1495463 - r1495444;
        double r1495465 = 2.0;
        double r1495466 = r1495464 / r1495465;
        double r1495467 = r1495466 / r1495449;
        double r1495468 = -1.0;
        double r1495469 = r1495468 * r1495448;
        double r1495470 = r1495455 ? r1495467 : r1495469;
        double r1495471 = r1495446 ? r1495453 : r1495470;
        return r1495471;
}

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 < -5.6179139475653e+116

    1. Initial program 52.0

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

    2. Simplified52.0

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

    3. Taylor expanded around -inf 3.7

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

    4. Simplified3.7

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

    if -5.6179139475653e+116 < b < 2.8983489306952693e-35

    1. Initial program 15.1

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

    2. Simplified15.0

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

    4. Applied add-sqr-sqrt15.0

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

    5. Applied sqrt-prod15.2

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))