Average Error: 34.7 → 8.5
Time: 37.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 123203260287366115360768:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 123203260287366115360768:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1810506 = b;
        double r1810507 = -r1810506;
        double r1810508 = r1810506 * r1810506;
        double r1810509 = 4.0;
        double r1810510 = a;
        double r1810511 = r1810509 * r1810510;
        double r1810512 = c;
        double r1810513 = r1810511 * r1810512;
        double r1810514 = r1810508 - r1810513;
        double r1810515 = sqrt(r1810514);
        double r1810516 = r1810507 + r1810515;
        double r1810517 = 2.0;
        double r1810518 = r1810517 * r1810510;
        double r1810519 = r1810516 / r1810518;
        return r1810519;
}
↓
double f(double a, double b, double c) {
        double r1810520 = b;
        double r1810521 = -1.2705286994550075e+152;
        bool r1810522 = r1810520 <= r1810521;
        double r1810523 = c;
        double r1810524 = r1810523 / r1810520;
        double r1810525 = a;
        double r1810526 = r1810520 / r1810525;
        double r1810527 = r1810524 - r1810526;
        double r1810528 = 1.0;
        double r1810529 = r1810527 * r1810528;
        double r1810530 = 2.1777947371956335e-143;
        bool r1810531 = r1810520 <= r1810530;
        double r1810532 = r1810520 * r1810520;
        double r1810533 = 4.0;
        double r1810534 = r1810523 * r1810533;
        double r1810535 = r1810534 * r1810525;
        double r1810536 = r1810532 - r1810535;
        double r1810537 = sqrt(r1810536);
        double r1810538 = r1810537 - r1810520;
        double r1810539 = 2.0;
        double r1810540 = r1810525 * r1810539;
        double r1810541 = r1810538 / r1810540;
        double r1810542 = 1.2320326028736612e+23;
        bool r1810543 = r1810520 <= r1810542;
        double r1810544 = r1810532 - r1810532;
        double r1810545 = r1810525 * r1810533;
        double r1810546 = r1810523 * r1810545;
        double r1810547 = r1810544 + r1810546;
        double r1810548 = -r1810520;
        double r1810549 = r1810532 - r1810546;
        double r1810550 = sqrt(r1810549);
        double r1810551 = r1810548 - r1810550;
        double r1810552 = r1810547 / r1810551;
        double r1810553 = r1810552 / r1810540;
        double r1810554 = -1.0;
        double r1810555 = r1810524 * r1810554;
        double r1810556 = r1810543 ? r1810553 : r1810555;
        double r1810557 = r1810531 ? r1810541 : r1810556;
        double r1810558 = r1810522 ? r1810529 : r1810557;
        return r1810558;
}

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 < -1.2705286994550075e+152

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 1.7

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

    3. Simplified1.7

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

    if -1.2705286994550075e+152 < b < 2.1777947371956335e-143

    1. Initial program 10.4

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

    3. Applied div-inv10.5

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

    5. Applied associate-*r/10.4

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

    6. Simplified10.4

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

    if 2.1777947371956335e-143 < b < 1.2320326028736612e+23

    1. Initial program 36.9

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

    3. Applied flip-+37.0

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

    4. Simplified18.3

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

    if 1.2320326028736612e+23 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 4.4

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 123203260287366115360768:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))