Average Error: 33.2 → 10.0
Time: 23.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -7.397994825724217 \cdot 10^{+150}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1901503 = b;
        double r1901504 = -r1901503;
        double r1901505 = r1901503 * r1901503;
        double r1901506 = 4.0;
        double r1901507 = a;
        double r1901508 = r1901506 * r1901507;
        double r1901509 = c;
        double r1901510 = r1901508 * r1901509;
        double r1901511 = r1901505 - r1901510;
        double r1901512 = sqrt(r1901511);
        double r1901513 = r1901504 + r1901512;
        double r1901514 = 2.0;
        double r1901515 = r1901514 * r1901507;
        double r1901516 = r1901513 / r1901515;
        return r1901516;
}
↓
double f(double a, double b, double c) {
        double r1901517 = b;
        double r1901518 = -7.397994825724217e+150;
        bool r1901519 = r1901517 <= r1901518;
        double r1901520 = c;
        double r1901521 = r1901520 / r1901517;
        double r1901522 = a;
        double r1901523 = r1901517 / r1901522;
        double r1901524 = r1901521 - r1901523;
        double r1901525 = 2.0;
        double r1901526 = r1901524 * r1901525;
        double r1901527 = r1901526 / r1901525;
        double r1901528 = 1.2158870426682226e-82;
        bool r1901529 = r1901517 <= r1901528;
        double r1901530 = 1.0;
        double r1901531 = r1901517 * r1901517;
        double r1901532 = r1901522 * r1901520;
        double r1901533 = 4.0;
        double r1901534 = r1901532 * r1901533;
        double r1901535 = r1901531 - r1901534;
        double r1901536 = sqrt(r1901535);
        double r1901537 = r1901522 / r1901536;
        double r1901538 = r1901530 / r1901537;
        double r1901539 = r1901538 - r1901523;
        double r1901540 = r1901539 / r1901525;
        double r1901541 = -2.0;
        double r1901542 = r1901541 * r1901521;
        double r1901543 = r1901542 / r1901525;
        double r1901544 = r1901529 ? r1901540 : r1901543;
        double r1901545 = r1901519 ? r1901527 : r1901544;
        return r1901545;
}

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 < -7.397994825724217e+150

    1. Initial program 59.1

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

    2. Simplified59.1

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -7.397994825724217e+150 < b < 1.2158870426682226e-82

    1. Initial program 11.8

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

    2. Simplified11.7

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

    4. Applied div-sub11.7

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

    6. Applied clear-num11.8

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

    if 1.2158870426682226e-82 < 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. Simplified52.3

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

    3. Taylor expanded around inf 9.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))