Average Error: 20.0 → 6.9
Time: 4.1s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -8.403599521210245 \cdot 10^{+110} \lor \neg \left(b \leq 2.1413501573873043 \cdot 10^{+105}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -8.403599521210245 \cdot 10^{+110} \lor \neg \left(b \leq 2.1413501573873043 \cdot 10^{+105}\right):\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a}\\

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

\end{array}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}\\

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

\end{array}
(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
   (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))
↓
(FPCore (a b c)
 :precision binary64
 (if (or (<= b -8.403599521210245e+110) (not (<= b 2.1413501573873043e+105)))
   (if (>= b 0.0)
     (* -0.5 (/ (+ b (- b (* 2.0 (/ a (/ b c))))) a))
     (/ (* 2.0 c) (* 2.0 (- (/ a (/ b c)) b))))
   (if (>= b 0.0)
     (*
      -0.5
      (/
       (+
        b
        (*
         (sqrt (sqrt (- (* b b) (* c (* a 4.0)))))
         (sqrt (sqrt (- (* b b) (* c (* a 4.0)))))))
       a))
     (/ (* 2.0 c) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))
double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + sqrt((b * b) - ((4.0 * a) * c)));
	}
	return tmp;
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= -8.403599521210245e+110) || !(b <= 2.1413501573873043e+105)) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = -0.5 * ((b + (b - (2.0 * (a / (b / c))))) / a);
		} else {
			tmp_1 = (2.0 * c) / (2.0 * ((a / (b / c)) - b));
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = -0.5 * ((b + (sqrt(sqrt((b * b) - (c * (a * 4.0)))) * sqrt(sqrt((b * b) - (c * (a * 4.0)))))) / a);
	} else {
		tmp = (2.0 * c) / (sqrt((b * b) - (c * (a * 4.0))) - b);
	}
	return tmp;
}

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 2 regimes

  2. if b < -8.40359952121024462e110 or 2.14135015738730429e105 < b

    1. Initial program 38.2

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

    2. Simplified38.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

    3. Taylor expanded around -inf 23.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

    4. Simplified23.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]
    5. Using strategy rm

    6. Applied associate-/l*_binary6421.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\color{blue}{2} \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \end{array}\]

    7. Taylor expanded around inf 5.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \end{array}\]

    8. Simplified2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\left(b - 2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \end{array}\]

    if -8.40359952121024462e110 < b < 2.14135015738730429e105

    1. Initial program 9.2

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

    2. Simplified9.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt_binary649.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

    5. Applied sqrt-prod_binary649.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.403599521210245 \cdot 10^{+110} \lor \neg \left(b \leq 2.1413501573873043 \cdot 10^{+105}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))