Average Error: 34.9 → 14.9
Time: 15.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361371441856741221428064888771237209188 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.678096336022247144956343717857518180364 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]
\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 -1.361371441856741221428064888771237209188 \cdot 10^{154}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\

\mathbf{elif}\;b \le 4.678096336022247144956343717857518180364 \cdot 10^{-33}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r43569 = b;
        double r43570 = -r43569;
        double r43571 = r43569 * r43569;
        double r43572 = 4.0;
        double r43573 = a;
        double r43574 = r43572 * r43573;
        double r43575 = c;
        double r43576 = r43574 * r43575;
        double r43577 = r43571 - r43576;
        double r43578 = sqrt(r43577);
        double r43579 = r43570 + r43578;
        double r43580 = 2.0;
        double r43581 = r43580 * r43573;
        double r43582 = r43579 / r43581;
        return r43582;
}
↓
double f(double a, double b, double c) {
        double r43583 = b;
        double r43584 = -1.3613714418567412e+154;
        bool r43585 = r43583 <= r43584;
        double r43586 = 2.0;
        double r43587 = a;
        double r43588 = c;
        double r43589 = r43587 * r43588;
        double r43590 = r43589 / r43583;
        double r43591 = r43586 * r43590;
        double r43592 = r43591 - r43583;
        double r43593 = r43592 - r43583;
        double r43594 = r43586 * r43587;
        double r43595 = r43593 / r43594;
        double r43596 = 4.678096336022247e-33;
        bool r43597 = r43583 <= r43596;
        double r43598 = r43583 * r43583;
        double r43599 = 4.0;
        double r43600 = r43599 * r43587;
        double r43601 = r43600 * r43588;
        double r43602 = r43598 - r43601;
        double r43603 = sqrt(r43602);
        double r43604 = r43603 - r43583;
        double r43605 = r43604 / r43594;
        double r43606 = -2.0;
        double r43607 = r43606 * r43590;
        double r43608 = r43607 / r43594;
        double r43609 = r43597 ? r43605 : r43608;
        double r43610 = r43585 ? r43595 : r43609;
        return r43610;
}

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 < -1.3613714418567412e+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

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

    3. Taylor expanded around -inf 10.0

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

    if -1.3613714418567412e+154 < b < 4.678096336022247e-33

    1. Initial program 14.3

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

    2. Simplified14.3

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

    if 4.678096336022247e-33 < b

    1. Initial program 55.5

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

    2. Simplified55.5

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

    3. Taylor expanded around inf 17.3

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

  4. Final simplification14.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361371441856741221428064888771237209188 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.678096336022247144956343717857518180364 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))