Average Error: 34.1 → 10.5
Time: 4.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\
\;\;\;\;\frac{1}{2} \cdot \left(\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r45753 = b;
        double r45754 = -r45753;
        double r45755 = r45753 * r45753;
        double r45756 = 4.0;
        double r45757 = a;
        double r45758 = c;
        double r45759 = r45757 * r45758;
        double r45760 = r45756 * r45759;
        double r45761 = r45755 - r45760;
        double r45762 = sqrt(r45761);
        double r45763 = r45754 - r45762;
        double r45764 = 2.0;
        double r45765 = r45764 * r45757;
        double r45766 = r45763 / r45765;
        return r45766;
}
↓
double f(double a, double b, double c) {
        double r45767 = b;
        double r45768 = -5.6439472304372656e-71;
        bool r45769 = r45767 <= r45768;
        double r45770 = -1.0;
        double r45771 = c;
        double r45772 = r45771 / r45767;
        double r45773 = r45770 * r45772;
        double r45774 = 1.4983382189642058e+54;
        bool r45775 = r45767 <= r45774;
        double r45776 = 1.0;
        double r45777 = 2.0;
        double r45778 = r45776 / r45777;
        double r45779 = a;
        double r45780 = r45776 / r45779;
        double r45781 = -r45767;
        double r45782 = r45767 * r45767;
        double r45783 = 4.0;
        double r45784 = r45779 * r45771;
        double r45785 = r45783 * r45784;
        double r45786 = r45782 - r45785;
        double r45787 = sqrt(r45786);
        double r45788 = r45781 - r45787;
        double r45789 = r45780 * r45788;
        double r45790 = r45778 * r45789;
        double r45791 = 1.0;
        double r45792 = r45767 / r45779;
        double r45793 = r45772 - r45792;
        double r45794 = r45791 * r45793;
        double r45795 = r45775 ? r45790 : r45794;
        double r45796 = r45769 ? r45773 : r45795;
        return r45796;
}

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
Target20.8
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.6439472304372656e-71

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 9.2

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

    if -5.6439472304372656e-71 < b < 1.4983382189642058e+54

    1. Initial program 14.3

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

    3. Applied *-un-lft-identity14.3

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

    4. Applied times-frac14.2

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

    6. Applied clear-num14.4

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

    8. Applied div-inv14.4

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

    9. Applied add-cube-cbrt14.4

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

    10. Applied times-frac14.4

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

    11. Simplified14.4

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

    12. Simplified14.4

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

    if 1.4983382189642058e+54 < b

    1. Initial program 37.9

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

    2. Taylor expanded around inf 5.0

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

    3. Simplified5.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

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