Average Error: 34.1 → 8.2
Time: 5.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\
\;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r59660 = b;
        double r59661 = -r59660;
        double r59662 = r59660 * r59660;
        double r59663 = 4.0;
        double r59664 = a;
        double r59665 = r59663 * r59664;
        double r59666 = c;
        double r59667 = r59665 * r59666;
        double r59668 = r59662 - r59667;
        double r59669 = sqrt(r59668);
        double r59670 = r59661 + r59669;
        double r59671 = 2.0;
        double r59672 = r59671 * r59664;
        double r59673 = r59670 / r59672;
        return r59673;
}
↓
double f(double a, double b, double c) {
        double r59674 = b;
        double r59675 = -3.974595954042362e+78;
        bool r59676 = r59674 <= r59675;
        double r59677 = 1.0;
        double r59678 = c;
        double r59679 = r59678 / r59674;
        double r59680 = a;
        double r59681 = r59674 / r59680;
        double r59682 = r59679 - r59681;
        double r59683 = r59677 * r59682;
        double r59684 = -3.106669992168921e-249;
        bool r59685 = r59674 <= r59684;
        double r59686 = -r59674;
        double r59687 = r59674 * r59674;
        double r59688 = 4.0;
        double r59689 = r59688 * r59680;
        double r59690 = r59689 * r59678;
        double r59691 = r59687 - r59690;
        double r59692 = sqrt(r59691);
        double r59693 = r59686 + r59692;
        double r59694 = 1.0;
        double r59695 = 2.0;
        double r59696 = r59695 * r59680;
        double r59697 = r59694 / r59696;
        double r59698 = r59693 * r59697;
        double r59699 = 3.3294485045805704e-14;
        bool r59700 = r59674 <= r59699;
        double r59701 = r59694 / r59680;
        double r59702 = r59686 - r59692;
        double r59703 = r59702 / r59678;
        double r59704 = r59701 * r59703;
        double r59705 = r59688 / r59704;
        double r59706 = r59694 * r59705;
        double r59707 = r59706 / r59696;
        double r59708 = -1.0;
        double r59709 = r59708 * r59679;
        double r59710 = r59700 ? r59707 : r59709;
        double r59711 = r59685 ? r59698 : r59710;
        double r59712 = r59676 ? r59683 : r59711;
        return r59712;
}

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 < -3.974595954042362e+78

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 4.2

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

    3. Simplified4.2

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

    if -3.974595954042362e+78 < b < -3.106669992168921e-249

    1. Initial program 8.2

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

    3. Applied div-inv8.3

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

    if -3.106669992168921e-249 < b < 3.3294485045805704e-14

    1. Initial program 23.6

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

    3. Applied flip-+23.7

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

    4. Simplified17.5

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity17.5

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

    7. Applied *-un-lft-identity17.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    8. Applied times-frac17.5

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    9. Simplified17.5

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

    10. Simplified17.5

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

    12. Applied *-un-lft-identity17.5

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

    13. Applied times-frac14.8

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

    if 3.3294485045805704e-14 < b

    1. Initial program 55.3

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

    2. Taylor expanded around inf 5.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))