Average Error: 34.1 → 9.9
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r70678 = b;
        double r70679 = -r70678;
        double r70680 = r70678 * r70678;
        double r70681 = 4.0;
        double r70682 = a;
        double r70683 = c;
        double r70684 = r70682 * r70683;
        double r70685 = r70681 * r70684;
        double r70686 = r70680 - r70685;
        double r70687 = sqrt(r70686);
        double r70688 = r70679 + r70687;
        double r70689 = 2.0;
        double r70690 = r70689 * r70682;
        double r70691 = r70688 / r70690;
        return r70691;
}
↓
double f(double a, double b, double c) {
        double r70692 = b;
        double r70693 = -1.5501620157466267e+150;
        bool r70694 = r70692 <= r70693;
        double r70695 = 1.0;
        double r70696 = c;
        double r70697 = r70696 / r70692;
        double r70698 = a;
        double r70699 = r70692 / r70698;
        double r70700 = r70697 - r70699;
        double r70701 = r70695 * r70700;
        double r70702 = 1.611450844781215e-34;
        bool r70703 = r70692 <= r70702;
        double r70704 = 1.0;
        double r70705 = 2.0;
        double r70706 = r70705 * r70698;
        double r70707 = r70692 * r70692;
        double r70708 = 4.0;
        double r70709 = r70698 * r70696;
        double r70710 = r70708 * r70709;
        double r70711 = r70707 - r70710;
        double r70712 = sqrt(r70711);
        double r70713 = r70712 - r70692;
        double r70714 = r70706 / r70713;
        double r70715 = r70704 / r70714;
        double r70716 = -1.0;
        double r70717 = r70716 * r70697;
        double r70718 = r70703 ? r70715 : r70717;
        double r70719 = r70694 ? r70701 : r70718;
        return r70719;
}

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

Target

Original34.1
Target21.2
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5501620157466267e+150

    1. Initial program 62.9

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

    2. Simplified62.9

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

    3. Taylor expanded around -inf 1.7

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

    4. Simplified1.7

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

    if -1.5501620157466267e+150 < b < 1.611450844781215e-34

    1. Initial program 13.6

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

    2. Simplified13.6

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

    4. Applied clear-num13.7

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

    if 1.611450844781215e-34 < b

    1. Initial program 55.0

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

    2. Simplified55.0

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

    3. Taylor expanded around inf 7.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))