Average Error: 20.0 → 6.7
Time: 5.7s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.57239956873832 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 3.2284907585376 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.4342762859357804 \cdot 10^{+50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b \cdot -2 + c \cdot \frac{2 \cdot a}{b}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -6.57239956873832 \cdot 10^{+148}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;2 \cdot \frac{-c}{b}\\

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

\end{array}\\

\mathbf{elif}\;b \leq 3.2284907585376 \cdot 10^{-310}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \leq 1.4342762859357804 \cdot 10^{+50}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\

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

\end{array}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;2 \cdot \frac{c}{b \cdot -2 + c \cdot \frac{2 \cdot a}{b}}\\

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

\end{array}
double code(double a, double b, double c) {
	double VAR;
	if ((b >= 0.0)) {
		VAR = (((double) (2.0 * c)) / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))));
	} else {
		VAR = (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
	}
	return VAR;
}
↓
double code(double a, double b, double c) {
	double VAR;
	if ((b <= -6.57239956873832e+148)) {
		double VAR_1;
		if ((b >= 0.0)) {
			VAR_1 = ((double) (2.0 * (((double) -(c)) / b)));
		} else {
			VAR_1 = ((double) (1.0 * ((double) ((c / b) - (b / a)))));
		}
		VAR = VAR_1;
	} else {
		double VAR_2;
		if ((b <= 3.2284907585376e-310)) {
			double VAR_3;
			if ((b >= 0.0)) {
				VAR_3 = ((double) (2.0 * (c / ((double) (((double) (((double) (2.0 * ((double) (c * (a / b))))) - b)) - b)))));
			} else {
				VAR_3 = (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))) - b)) / ((double) (2.0 * a)));
			}
			VAR_2 = VAR_3;
		} else {
			double VAR_4;
			if ((b <= 1.4342762859357804e+50)) {
				double VAR_5;
				if ((b >= 0.0)) {
					VAR_5 = ((double) (2.0 * (c / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))))))));
				} else {
					VAR_5 = (((double) (((double) (2.0 * (((double) (c * a)) / b))) + ((double) (b * -2.0)))) / ((double) (2.0 * a)));
				}
				VAR_4 = VAR_5;
			} else {
				double VAR_6;
				if ((b >= 0.0)) {
					VAR_6 = ((double) (2.0 * (c / ((double) (((double) (b * -2.0)) + ((double) (c * (((double) (2.0 * a)) / b))))))));
				} else {
					VAR_6 = ((double) (1.0 * ((double) ((c / b) - (b / a)))));
				}
				VAR_4 = VAR_6;
			}
			VAR_2 = VAR_4;
		}
		VAR = VAR_2;
	}
	return VAR;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -6.5723995687383196e148

    1. Initial program 61.0

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

    2. Simplified61.0

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around -inf 10.1

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

    4. Simplified2.5

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

    5. Taylor expanded around 0 2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\\ \end{array}\]

    6. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    7. Taylor expanded around inf 2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \color{blue}{\left(-1 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    8. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    if -6.5723995687383196e148 < b < 3.22849075853761e-310

    1. Initial program 8.4

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

    2. Simplified8.4

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around inf 8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\]

    4. Simplified8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\]

    if 3.22849075853761e-310 < b < 1.4342762859357804e50

    1. Initial program 9.2

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

    2. Simplified9.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around -inf 9.2

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

    4. Simplified9.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{2 \cdot a}\\ \end{array}\]
    5. Using strategy rm

    6. Applied associate-*r/9.2

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

    if 1.4342762859357804e50 < b

    1. Initial program 26.3

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

    2. Simplified26.3

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around -inf 26.3

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

    4. Simplified26.3

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

    5. Taylor expanded around 0 26.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\\ \end{array}\]

    6. Simplified26.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    7. Taylor expanded around inf 7.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

    8. Simplified4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{c \cdot \frac{a \cdot 2}{b} + b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.57239956873832 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 3.2284907585376 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.4342762859357804 \cdot 10^{+50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b \cdot -2 + c \cdot \frac{2 \cdot a}{b}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))