Average Error: 34.1 → 10.5
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r99744 = b;
        double r99745 = -r99744;
        double r99746 = r99744 * r99744;
        double r99747 = 4.0;
        double r99748 = a;
        double r99749 = c;
        double r99750 = r99748 * r99749;
        double r99751 = r99747 * r99750;
        double r99752 = r99746 - r99751;
        double r99753 = sqrt(r99752);
        double r99754 = r99745 - r99753;
        double r99755 = 2.0;
        double r99756 = r99755 * r99748;
        double r99757 = r99754 / r99756;
        return r99757;
}
↓
double f(double a, double b, double c) {
        double r99758 = b;
        double r99759 = -5.6439472304372656e-71;
        bool r99760 = r99758 <= r99759;
        double r99761 = -1.0;
        double r99762 = c;
        double r99763 = r99762 / r99758;
        double r99764 = r99761 * r99763;
        double r99765 = 1.4983382189642058e+54;
        bool r99766 = r99758 <= r99765;
        double r99767 = 1.0;
        double r99768 = 2.0;
        double r99769 = r99767 / r99768;
        double r99770 = -r99758;
        double r99771 = r99758 * r99758;
        double r99772 = 4.0;
        double r99773 = a;
        double r99774 = r99773 * r99762;
        double r99775 = r99772 * r99774;
        double r99776 = r99771 - r99775;
        double r99777 = sqrt(r99776);
        double r99778 = r99770 - r99777;
        double r99779 = r99778 / r99773;
        double r99780 = r99769 * r99779;
        double r99781 = 1.0;
        double r99782 = r99758 / r99773;
        double r99783 = r99763 - r99782;
        double r99784 = r99781 * r99783;
        double r99785 = r99766 ? r99780 : r99784;
        double r99786 = r99760 ? r99764 : r99785;
        return r99786;
}

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

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

    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

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

Reproduce

herbie shell --seed 2020002 
(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)))