Average Error: 34.3 → 10.0
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.66563711558993385 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.1585291365273219 \cdot 10^{122}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -9.66563711558993385 \cdot 10^{-69}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.1585291365273219 \cdot 10^{122}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r85677 = b;
        double r85678 = -r85677;
        double r85679 = r85677 * r85677;
        double r85680 = 4.0;
        double r85681 = a;
        double r85682 = c;
        double r85683 = r85681 * r85682;
        double r85684 = r85680 * r85683;
        double r85685 = r85679 - r85684;
        double r85686 = sqrt(r85685);
        double r85687 = r85678 - r85686;
        double r85688 = 2.0;
        double r85689 = r85688 * r85681;
        double r85690 = r85687 / r85689;
        return r85690;
}
↓
double f(double a, double b, double c) {
        double r85691 = b;
        double r85692 = -9.665637115589934e-69;
        bool r85693 = r85691 <= r85692;
        double r85694 = -1.0;
        double r85695 = c;
        double r85696 = r85695 / r85691;
        double r85697 = r85694 * r85696;
        double r85698 = 9.158529136527322e+122;
        bool r85699 = r85691 <= r85698;
        double r85700 = 1.0;
        double r85701 = 2.0;
        double r85702 = a;
        double r85703 = r85701 * r85702;
        double r85704 = -r85691;
        double r85705 = r85691 * r85691;
        double r85706 = 4.0;
        double r85707 = r85702 * r85695;
        double r85708 = r85706 * r85707;
        double r85709 = r85705 - r85708;
        double r85710 = sqrt(r85709);
        double r85711 = r85704 - r85710;
        double r85712 = r85703 / r85711;
        double r85713 = r85700 / r85712;
        double r85714 = r85691 / r85702;
        double r85715 = r85694 * r85714;
        double r85716 = r85699 ? r85713 : r85715;
        double r85717 = r85693 ? r85697 : r85716;
        return r85717;
}

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
Target21.3
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.665637115589934e-69

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 8.6

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

    if -9.665637115589934e-69 < b < 9.158529136527322e+122

    1. Initial program 13.0

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

    3. Applied clear-num13.2

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

    if 9.158529136527322e+122 < b

    1. Initial program 53.1

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

    3. Applied clear-num53.2

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

    4. Taylor expanded around 0 3.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.66563711558993385 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.1585291365273219 \cdot 10^{122}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))