Average Error: 20.0 → 8.6
Time: 5.6s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -425882.49394125835:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6.4417367273105074 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c \cdot a}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\]
\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 -425882.49394125835:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\

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

\end{array}\\

\mathbf{elif}\;b \leq 6.4417367273105074 \cdot 10^{+103}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\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}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}\\

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

\end{array}
(FPCore (a b c)
 :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))))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -425882.49394125835)
   (if (>= b 0.0)
     (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* c (* 4.0 a))))))
     (- (/ c b) (/ b a)))
   (if (<= b 6.4417367273105074e+103)
     (if (>= b 0.0)
       (/
        (* 2.0 c)
        (-
         (- b)
         (*
          (sqrt (sqrt (- (* b b) (* c (* 4.0 a)))))
          (sqrt (sqrt (- (* b b) (* c (* 4.0 a))))))))
       (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a)))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* 2.0 (- (/ (* c a) b) b)))
       (/ (- (- (* 2.0 (/ (* c a) b)) b) b) (* 2.0 a))))))
double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - sqrt((b * b) - ((4.0 * a) * c)));
	} else {
		tmp = (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
	}
	return tmp;
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -425882.49394125835) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (2.0 * c) / (-b - sqrt((b * b) - (c * (4.0 * a))));
		} else {
			tmp_1 = (c / b) - (b / a);
		}
		tmp = tmp_1;
	} else if (b <= 6.4417367273105074e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - (sqrt(sqrt((b * b) - (c * (4.0 * a)))) * sqrt(sqrt((b * b) - (c * (4.0 * a))))));
		} else {
			tmp_2 = (sqrt((b * b) - (c * (4.0 * a))) - b) / (2.0 * a);
		}
		tmp = tmp_2;
	} else if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * (((c * a) / b) - b));
	} else {
		tmp = (((2.0 * ((c * a) / b)) - b) - b) / (2.0 * a);
	}
	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 3 regimes

  2. if b < -425882.493941258348

    1. Initial program 32.2

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

    2. Taylor expanded around -inf 11.5

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

    3. Taylor expanded around 0 7.4

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

    if -425882.493941258348 < b < 6.441736727310507e103

    1. Initial program 9.8

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

    3. Applied add-sqr-sqrt_binary6410.0

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

    if 6.441736727310507e103 < b

    1. Initial program 30.5

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

    2. Taylor expanded around -inf 30.5

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

    3. Taylor expanded around inf 6.7

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

    4. Simplified6.7

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -425882.49394125835:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6.4417367273105074 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \sqrt{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c \cdot a}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))))