Average Error: 33.9 → 10.3
Time: 20.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r59738 = b;
        double r59739 = -r59738;
        double r59740 = r59738 * r59738;
        double r59741 = 4.0;
        double r59742 = a;
        double r59743 = c;
        double r59744 = r59742 * r59743;
        double r59745 = r59741 * r59744;
        double r59746 = r59740 - r59745;
        double r59747 = sqrt(r59746);
        double r59748 = r59739 + r59747;
        double r59749 = 2.0;
        double r59750 = r59749 * r59742;
        double r59751 = r59748 / r59750;
        return r59751;
}
↓
double f(double a, double b, double c) {
        double r59752 = b;
        double r59753 = -1.361733299857302e+105;
        bool r59754 = r59752 <= r59753;
        double r59755 = 1.0;
        double r59756 = c;
        double r59757 = r59756 / r59752;
        double r59758 = a;
        double r59759 = r59752 / r59758;
        double r59760 = r59757 - r59759;
        double r59761 = r59755 * r59760;
        double r59762 = 3.091361180800597e-86;
        bool r59763 = r59752 <= r59762;
        double r59764 = r59752 * r59752;
        double r59765 = 4.0;
        double r59766 = r59765 * r59756;
        double r59767 = r59758 * r59766;
        double r59768 = r59764 - r59767;
        double r59769 = sqrt(r59768);
        double r59770 = -r59752;
        double r59771 = r59769 + r59770;
        double r59772 = 2.0;
        double r59773 = r59772 * r59758;
        double r59774 = r59771 / r59773;
        double r59775 = -1.0;
        double r59776 = r59775 * r59757;
        double r59777 = r59763 ? r59774 : r59776;
        double r59778 = r59754 ? r59761 : r59777;
        return r59778;
}

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

Original33.9
Target21.1
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.361733299857302e+105 < b < 3.091361180800597e-86

    1. Initial program 12.2

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

    3. Applied *-un-lft-identity12.2

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

    4. Applied associate-*l*12.2

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

    5. Simplified12.2

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

    if 3.091361180800597e-86 < b

    1. Initial program 51.8

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))