Average Error: 33.7 → 8.3
Time: 17.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.234164035284793 \cdot 10^{+22}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.3209183644448 \cdot 10^{-115}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\

\mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3523707 = b;
        double r3523708 = -r3523707;
        double r3523709 = r3523707 * r3523707;
        double r3523710 = 4.0;
        double r3523711 = a;
        double r3523712 = c;
        double r3523713 = r3523711 * r3523712;
        double r3523714 = r3523710 * r3523713;
        double r3523715 = r3523709 - r3523714;
        double r3523716 = sqrt(r3523715);
        double r3523717 = r3523708 - r3523716;
        double r3523718 = 2.0;
        double r3523719 = r3523718 * r3523711;
        double r3523720 = r3523717 / r3523719;
        return r3523720;
}
↓
double f(double a, double b, double c) {
        double r3523721 = b;
        double r3523722 = -3.234164035284793e+22;
        bool r3523723 = r3523721 <= r3523722;
        double r3523724 = c;
        double r3523725 = r3523724 / r3523721;
        double r3523726 = -r3523725;
        double r3523727 = -6.3209183644448e-115;
        bool r3523728 = r3523721 <= r3523727;
        double r3523729 = 0.5;
        double r3523730 = a;
        double r3523731 = r3523729 / r3523730;
        double r3523732 = r3523730 * r3523724;
        double r3523733 = 4.0;
        double r3523734 = r3523732 * r3523733;
        double r3523735 = r3523721 * r3523721;
        double r3523736 = r3523735 - r3523734;
        double r3523737 = sqrt(r3523736);
        double r3523738 = r3523737 - r3523721;
        double r3523739 = r3523734 / r3523738;
        double r3523740 = r3523731 * r3523739;
        double r3523741 = 2.026128983134594e+103;
        bool r3523742 = r3523721 <= r3523741;
        double r3523743 = -r3523721;
        double r3523744 = r3523743 - r3523737;
        double r3523745 = r3523729 * r3523744;
        double r3523746 = r3523745 / r3523730;
        double r3523747 = r3523721 / r3523730;
        double r3523748 = r3523725 - r3523747;
        double r3523749 = r3523742 ? r3523746 : r3523748;
        double r3523750 = r3523728 ? r3523740 : r3523749;
        double r3523751 = r3523723 ? r3523726 : r3523750;
        return r3523751;
}

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

Original33.7
Target20.9
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.234164035284793e+22

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 4.6

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

    3. Simplified4.6

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

    if -3.234164035284793e+22 < b < -6.3209183644448e-115

    1. Initial program 38.4

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

    3. Applied div-inv38.4

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

    4. Simplified38.4

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

    6. Applied flip--38.5

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

    7. Simplified15.7

      \[\leadsto \frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{\frac{1}{2}}{a}\]

    8. Simplified15.7

      \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\frac{1}{2}}{a}\]

    if -6.3209183644448e-115 < b < 2.026128983134594e+103

    1. Initial program 11.3

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

    3. Applied div-inv11.4

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

    4. Simplified11.4

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

    6. Applied associate-*r/11.3

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

    if 2.026128983134594e+103 < b

    1. Initial program 45.2

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "quadm (p42, negative)"

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