Average Error: 34.2 → 14.6
Time: 18.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.339237199229079299889677574983198023934 \cdot 10^{154}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{a \cdot c}{b} - b\right)}{a}\\ \mathbf{elif}\;b \le 1.915204981423677423459982128341604006799 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]
\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 -1.339237199229079299889677574983198023934 \cdot 10^{154}:\\
\;\;\;\;\frac{1 \cdot \left(\frac{a \cdot c}{b} - b\right)}{a}\\

\mathbf{elif}\;b \le 1.915204981423677423459982128341604006799 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2799635 = b;
        double r2799636 = -r2799635;
        double r2799637 = r2799635 * r2799635;
        double r2799638 = 4.0;
        double r2799639 = a;
        double r2799640 = r2799638 * r2799639;
        double r2799641 = c;
        double r2799642 = r2799640 * r2799641;
        double r2799643 = r2799637 - r2799642;
        double r2799644 = sqrt(r2799643);
        double r2799645 = r2799636 + r2799644;
        double r2799646 = 2.0;
        double r2799647 = r2799646 * r2799639;
        double r2799648 = r2799645 / r2799647;
        return r2799648;
}
↓
double f(double a, double b, double c) {
        double r2799649 = b;
        double r2799650 = -1.3392371992290793e+154;
        bool r2799651 = r2799649 <= r2799650;
        double r2799652 = 1.0;
        double r2799653 = a;
        double r2799654 = c;
        double r2799655 = r2799653 * r2799654;
        double r2799656 = r2799655 / r2799649;
        double r2799657 = r2799656 - r2799649;
        double r2799658 = r2799652 * r2799657;
        double r2799659 = r2799658 / r2799653;
        double r2799660 = 1.9152049814236774e-46;
        bool r2799661 = r2799649 <= r2799660;
        double r2799662 = r2799649 * r2799649;
        double r2799663 = 4.0;
        double r2799664 = r2799663 * r2799653;
        double r2799665 = r2799654 * r2799664;
        double r2799666 = r2799662 - r2799665;
        double r2799667 = sqrt(r2799666);
        double r2799668 = r2799667 - r2799649;
        double r2799669 = 2.0;
        double r2799670 = r2799668 / r2799669;
        double r2799671 = r2799670 / r2799653;
        double r2799672 = -1.0;
        double r2799673 = r2799672 * r2799656;
        double r2799674 = r2799673 / r2799653;
        double r2799675 = r2799661 ? r2799671 : r2799674;
        double r2799676 = r2799651 ? r2799659 : r2799675;
        return r2799676;
}

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 < -1.3392371992290793e+154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 9.5

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

    4. Simplified9.5

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

    if -1.3392371992290793e+154 < b < 1.9152049814236774e-46

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied add-cube-cbrt13.1

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

    5. Applied associate-*r*13.1

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

    7. Applied *-un-lft-identity13.1

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{1 \cdot \left(b \cdot b - \left(\left(4 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}\right)}} - b}{2}}{a}\]

    8. Applied sqrt-prod13.1

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{b \cdot b - \left(\left(4 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}}} - b}{2}}{a}\]

    9. Simplified13.1

      \[\leadsto \frac{\frac{\color{blue}{1} \cdot \sqrt{b \cdot b - \left(\left(4 \cdot a\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}} - b}{2}}{a}\]

    10. Simplified12.8

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

    if 1.9152049814236774e-46 < b

    1. Initial program 54.2

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

    2. Simplified54.2

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

    3. Taylor expanded around inf 18.5

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

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.339237199229079299889677574983198023934 \cdot 10^{154}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{a \cdot c}{b} - b\right)}{a}\\ \mathbf{elif}\;b \le 1.915204981423677423459982128341604006799 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot c}{b}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))