Average Error: 34.3 → 9.2
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.850740894150185 \cdot 10^{20}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.1806208921043762 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.29361288009581358 \cdot 10^{112}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.850740894150185 \cdot 10^{20}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -6.1806208921043762 \cdot 10^{-95}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le -2.125553485370055 \cdot 10^{-113}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.29361288009581358 \cdot 10^{112}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r73742 = b;
        double r73743 = -r73742;
        double r73744 = r73742 * r73742;
        double r73745 = 4.0;
        double r73746 = a;
        double r73747 = c;
        double r73748 = r73746 * r73747;
        double r73749 = r73745 * r73748;
        double r73750 = r73744 - r73749;
        double r73751 = sqrt(r73750);
        double r73752 = r73743 - r73751;
        double r73753 = 2.0;
        double r73754 = r73753 * r73746;
        double r73755 = r73752 / r73754;
        return r73755;
}
↓
double f(double a, double b, double c) {
        double r73756 = b;
        double r73757 = -1.8507408941501855e+20;
        bool r73758 = r73756 <= r73757;
        double r73759 = -1.0;
        double r73760 = c;
        double r73761 = r73760 / r73756;
        double r73762 = r73759 * r73761;
        double r73763 = -6.180620892104376e-95;
        bool r73764 = r73756 <= r73763;
        double r73765 = 1.0;
        double r73766 = 2.0;
        double r73767 = r73765 / r73766;
        double r73768 = 2.0;
        double r73769 = pow(r73756, r73768);
        double r73770 = r73769 - r73769;
        double r73771 = 4.0;
        double r73772 = a;
        double r73773 = r73772 * r73760;
        double r73774 = r73771 * r73773;
        double r73775 = r73770 + r73774;
        double r73776 = r73767 * r73775;
        double r73777 = r73776 / r73772;
        double r73778 = -r73756;
        double r73779 = r73756 * r73756;
        double r73780 = r73779 - r73774;
        double r73781 = sqrt(r73780);
        double r73782 = r73778 + r73781;
        double r73783 = r73765 / r73782;
        double r73784 = r73777 * r73783;
        double r73785 = -2.125553485370055e-113;
        bool r73786 = r73756 <= r73785;
        double r73787 = 6.293612880095814e+112;
        bool r73788 = r73756 <= r73787;
        double r73789 = r73778 - r73781;
        double r73790 = r73766 * r73772;
        double r73791 = r73765 / r73790;
        double r73792 = r73789 * r73791;
        double r73793 = r73756 / r73772;
        double r73794 = r73759 * r73793;
        double r73795 = r73788 ? r73792 : r73794;
        double r73796 = r73786 ? r73762 : r73795;
        double r73797 = r73764 ? r73784 : r73796;
        double r73798 = r73758 ? r73762 : r73797;
        return r73798;
}

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

Original34.3
Target21.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8507408941501855e+20 or -6.180620892104376e-95 < b < -2.125553485370055e-113

    1. Initial program 54.9

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

    2. Taylor expanded around -inf 6.3

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

    if -1.8507408941501855e+20 < b < -6.180620892104376e-95

    1. Initial program 39.1

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

    3. Applied clear-num39.1

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

    5. Applied flip--39.1

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

    6. Applied associate-/r/39.2

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

    7. Applied add-cube-cbrt39.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    8. Applied times-frac39.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{\sqrt[3]{1}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified14.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}} \cdot \frac{\sqrt[3]{1}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

    10. Simplified14.4

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

    if -2.125553485370055e-113 < b < 6.293612880095814e+112

    1. Initial program 12.2

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

    3. Applied div-inv12.3

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

    if 6.293612880095814e+112 < b

    1. Initial program 49.8

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

    3. Applied clear-num49.8

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

    4. Taylor expanded around 0 2.9

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.850740894150185 \cdot 10^{20}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.1806208921043762 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.29361288009581358 \cdot 10^{112}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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