Average Error: 34.4 → 10.4
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.566271626824697176623188586088934813649 \cdot 10^{80}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.053820360384825181255556663765142271045 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.566271626824697176623188586088934813649 \cdot 10^{80}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.053820360384825181255556663765142271045 \cdot 10^{-79}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r88765 = b;
        double r88766 = -r88765;
        double r88767 = r88765 * r88765;
        double r88768 = 4.0;
        double r88769 = a;
        double r88770 = c;
        double r88771 = r88769 * r88770;
        double r88772 = r88768 * r88771;
        double r88773 = r88767 - r88772;
        double r88774 = sqrt(r88773);
        double r88775 = r88766 + r88774;
        double r88776 = 2.0;
        double r88777 = r88776 * r88769;
        double r88778 = r88775 / r88777;
        return r88778;
}
↓
double f(double a, double b, double c) {
        double r88779 = b;
        double r88780 = -1.5662716268246972e+80;
        bool r88781 = r88779 <= r88780;
        double r88782 = 1.0;
        double r88783 = c;
        double r88784 = r88783 / r88779;
        double r88785 = a;
        double r88786 = r88779 / r88785;
        double r88787 = r88784 - r88786;
        double r88788 = r88782 * r88787;
        double r88789 = 4.053820360384825e-79;
        bool r88790 = r88779 <= r88789;
        double r88791 = -r88779;
        double r88792 = r88779 * r88779;
        double r88793 = 4.0;
        double r88794 = r88785 * r88783;
        double r88795 = r88793 * r88794;
        double r88796 = r88792 - r88795;
        double r88797 = sqrt(r88796);
        double r88798 = r88791 + r88797;
        double r88799 = sqrt(r88798);
        double r88800 = 2.0;
        double r88801 = r88799 / r88800;
        double r88802 = r88799 / r88785;
        double r88803 = r88801 * r88802;
        double r88804 = -1.0;
        double r88805 = r88804 * r88784;
        double r88806 = r88790 ? r88803 : r88805;
        double r88807 = r88781 ? r88788 : r88806;
        return r88807;
}

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

Original34.4
Target21.0
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5662716268246972e+80

    1. Initial program 41.9

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

    2. Taylor expanded around -inf 4.7

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

    3. Simplified4.7

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

    if -1.5662716268246972e+80 < b < 4.053820360384825e-79

    1. Initial program 13.4

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

    3. Applied add-sqr-sqrt13.8

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

    4. Applied times-frac13.8

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

    if 4.053820360384825e-79 < b

    1. Initial program 53.4

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.566271626824697176623188586088934813649 \cdot 10^{80}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.053820360384825181255556663765142271045 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(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)))