Average Error: 34.4 → 6.7
Time: 20.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r3806741 = b;
        double r3806742 = -r3806741;
        double r3806743 = r3806741 * r3806741;
        double r3806744 = 4.0;
        double r3806745 = a;
        double r3806746 = c;
        double r3806747 = r3806745 * r3806746;
        double r3806748 = r3806744 * r3806747;
        double r3806749 = r3806743 - r3806748;
        double r3806750 = sqrt(r3806749);
        double r3806751 = r3806742 - r3806750;
        double r3806752 = 2.0;
        double r3806753 = r3806752 * r3806745;
        double r3806754 = r3806751 / r3806753;
        return r3806754;
}
↓
double f(double a, double b, double c) {
        double r3806755 = b;
        double r3806756 = -1.7633154797394035e+89;
        bool r3806757 = r3806755 <= r3806756;
        double r3806758 = -1.0;
        double r3806759 = c;
        double r3806760 = r3806759 / r3806755;
        double r3806761 = r3806758 * r3806760;
        double r3806762 = -1.0850002786366243e-297;
        bool r3806763 = r3806755 <= r3806762;
        double r3806764 = 2.0;
        double r3806765 = r3806759 * r3806764;
        double r3806766 = -r3806755;
        double r3806767 = r3806755 * r3806755;
        double r3806768 = 4.0;
        double r3806769 = a;
        double r3806770 = r3806769 * r3806759;
        double r3806771 = r3806768 * r3806770;
        double r3806772 = r3806767 - r3806771;
        double r3806773 = sqrt(r3806772);
        double r3806774 = r3806766 + r3806773;
        double r3806775 = r3806765 / r3806774;
        double r3806776 = 3.355858625783055e+101;
        bool r3806777 = r3806755 <= r3806776;
        double r3806778 = r3806766 - r3806773;
        double r3806779 = r3806769 * r3806764;
        double r3806780 = r3806778 / r3806779;
        double r3806781 = r3806755 / r3806769;
        double r3806782 = r3806760 - r3806781;
        double r3806783 = 1.0;
        double r3806784 = r3806782 * r3806783;
        double r3806785 = r3806777 ? r3806780 : r3806784;
        double r3806786 = r3806763 ? r3806775 : r3806785;
        double r3806787 = r3806757 ? r3806761 : r3806786;
        return r3806787;
}

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
Target20.9
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 2.7

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

    if -1.7633154797394035e+89 < b < -1.0850002786366243e-297

    1. Initial program 32.1

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

    3. Applied div-inv32.1

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

    5. Applied flip--32.2

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

    6. Applied associate-*l/32.2

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

    7. Simplified15.8

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

    8. Taylor expanded around 0 8.4

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

    if -1.0850002786366243e-297 < b < 3.355858625783055e+101

    1. Initial program 9.5

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

    3. Applied div-inv9.6

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

    5. Applied un-div-inv9.5

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

    if 3.355858625783055e+101 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.085000278636624341855070450537604684134 \cdot 10^{-297}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))