Average Error: 34.1 → 10.0
Time: 18.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.644409376808577592769878215530106196344 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.542951486580225494391623071663800816645 \cdot 10^{93}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -2.644409376808577592769878215530106196344 \cdot 10^{-69}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.542951486580225494391623071663800816645 \cdot 10^{93}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r63735 = b;
        double r63736 = -r63735;
        double r63737 = r63735 * r63735;
        double r63738 = 4.0;
        double r63739 = a;
        double r63740 = c;
        double r63741 = r63739 * r63740;
        double r63742 = r63738 * r63741;
        double r63743 = r63737 - r63742;
        double r63744 = sqrt(r63743);
        double r63745 = r63736 - r63744;
        double r63746 = 2.0;
        double r63747 = r63746 * r63739;
        double r63748 = r63745 / r63747;
        return r63748;
}
↓
double f(double a, double b, double c) {
        double r63749 = b;
        double r63750 = -2.6444093768085776e-69;
        bool r63751 = r63749 <= r63750;
        double r63752 = -1.0;
        double r63753 = c;
        double r63754 = r63753 / r63749;
        double r63755 = r63752 * r63754;
        double r63756 = 2.5429514865802255e+93;
        bool r63757 = r63749 <= r63756;
        double r63758 = -r63749;
        double r63759 = r63749 * r63749;
        double r63760 = 4.0;
        double r63761 = a;
        double r63762 = r63761 * r63753;
        double r63763 = r63760 * r63762;
        double r63764 = r63759 - r63763;
        double r63765 = sqrt(r63764);
        double r63766 = r63758 - r63765;
        double r63767 = 2.0;
        double r63768 = r63766 / r63767;
        double r63769 = r63768 / r63761;
        double r63770 = -2.0;
        double r63771 = r63770 * r63749;
        double r63772 = r63767 * r63761;
        double r63773 = r63771 / r63772;
        double r63774 = r63757 ? r63769 : r63773;
        double r63775 = r63751 ? r63755 : r63774;
        return r63775;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -2.6444093768085776e-69

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 8.8

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

    if -2.6444093768085776e-69 < b < 2.5429514865802255e+93

    1. Initial program 13.4

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

    3. Applied associate-/r*13.4

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

    if 2.5429514865802255e+93 < 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. Using strategy rm

    3. Applied flip--62.9

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

    4. Simplified62.0

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

    5. Simplified62.0

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

    6. Taylor expanded around 0 3.5

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.644409376808577592769878215530106196344 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.542951486580225494391623071663800816645 \cdot 10^{93}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))