Average Error: 34.9 → 14.9
Time: 15.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361371441856741221428064888771237209188 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.678096336022247144956343717857518180364 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.361371441856741221428064888771237209188 \cdot 10^{154}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\

\mathbf{elif}\;b \le 4.678096336022247144956343717857518180364 \cdot 10^{-33}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r61658 = b;
        double r61659 = -r61658;
        double r61660 = r61658 * r61658;
        double r61661 = 4.0;
        double r61662 = a;
        double r61663 = r61661 * r61662;
        double r61664 = c;
        double r61665 = r61663 * r61664;
        double r61666 = r61660 - r61665;
        double r61667 = sqrt(r61666);
        double r61668 = r61659 + r61667;
        double r61669 = 2.0;
        double r61670 = r61669 * r61662;
        double r61671 = r61668 / r61670;
        return r61671;
}
↓
double f(double a, double b, double c) {
        double r61672 = b;
        double r61673 = -1.3613714418567412e+154;
        bool r61674 = r61672 <= r61673;
        double r61675 = 2.0;
        double r61676 = a;
        double r61677 = c;
        double r61678 = r61676 * r61677;
        double r61679 = r61678 / r61672;
        double r61680 = r61675 * r61679;
        double r61681 = r61680 - r61672;
        double r61682 = r61681 - r61672;
        double r61683 = r61675 * r61676;
        double r61684 = r61682 / r61683;
        double r61685 = 4.678096336022247e-33;
        bool r61686 = r61672 <= r61685;
        double r61687 = r61672 * r61672;
        double r61688 = 4.0;
        double r61689 = r61688 * r61676;
        double r61690 = r61689 * r61677;
        double r61691 = r61687 - r61690;
        double r61692 = sqrt(r61691);
        double r61693 = r61692 - r61672;
        double r61694 = r61693 / r61683;
        double r61695 = -2.0;
        double r61696 = r61695 * r61679;
        double r61697 = r61696 / r61683;
        double r61698 = r61686 ? r61694 : r61697;
        double r61699 = r61674 ? r61684 : r61698;
        return r61699;
}

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

Original34.9
Target21.3
Herbie14.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3613714418567412e+154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 10.0

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

    if -1.3613714418567412e+154 < b < 4.678096336022247e-33

    1. Initial program 14.3

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

    2. Simplified14.3

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

    if 4.678096336022247e-33 < b

    1. Initial program 55.5

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

    2. Simplified55.5

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

    3. Taylor expanded around inf 17.3

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

  4. Final simplification14.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361371441856741221428064888771237209188 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.678096336022247144956343717857518180364 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))