Average Error: 33.0 → 10.6
Time: 19.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.794505329565205 \cdot 10^{+146}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.6194276288860963:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2362745 = b;
        double r2362746 = -r2362745;
        double r2362747 = r2362745 * r2362745;
        double r2362748 = 4.0;
        double r2362749 = a;
        double r2362750 = c;
        double r2362751 = r2362749 * r2362750;
        double r2362752 = r2362748 * r2362751;
        double r2362753 = r2362747 - r2362752;
        double r2362754 = sqrt(r2362753);
        double r2362755 = r2362746 + r2362754;
        double r2362756 = 2.0;
        double r2362757 = r2362756 * r2362749;
        double r2362758 = r2362755 / r2362757;
        return r2362758;
}
↓
double f(double a, double b, double c) {
        double r2362759 = b;
        double r2362760 = -3.794505329565205e+146;
        bool r2362761 = r2362759 <= r2362760;
        double r2362762 = c;
        double r2362763 = r2362762 / r2362759;
        double r2362764 = a;
        double r2362765 = r2362759 / r2362764;
        double r2362766 = r2362763 - r2362765;
        double r2362767 = 1.6194276288860963;
        bool r2362768 = r2362759 <= r2362767;
        double r2362769 = r2362759 * r2362759;
        double r2362770 = 4.0;
        double r2362771 = r2362764 * r2362770;
        double r2362772 = r2362771 * r2362762;
        double r2362773 = r2362769 - r2362772;
        double r2362774 = sqrt(r2362773);
        double r2362775 = r2362774 - r2362759;
        double r2362776 = 2.0;
        double r2362777 = r2362764 * r2362776;
        double r2362778 = r2362775 / r2362777;
        double r2362779 = -r2362763;
        double r2362780 = r2362768 ? r2362778 : r2362779;
        double r2362781 = r2362761 ? r2362766 : r2362780;
        return r2362781;
}

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

Original33.0
Target20.2
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.794505329565205e+146

    1. Initial program 58.0

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

    2. Simplified58.0

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

    3. Taylor expanded around -inf 3.1

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

    if -3.794505329565205e+146 < b < 1.6194276288860963

    1. Initial program 15.0

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

    2. Simplified15.0

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

    3. Taylor expanded around -inf 15.0

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

    4. Simplified15.0

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

    if 1.6194276288860963 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    4. Applied div-inv54.5

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

    5. Simplified54.5

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

    6. Taylor expanded around inf 5.9

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

    7. Simplified5.9

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))