Average Error: 19.6 → 6.4
Time: 7.6s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
↓
\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\ t_1 := -\frac{c}{b}\\ t_2 := \left(-b\right) - t_0\\ \mathbf{if}\;b \leq -3.96398836197757 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 1.422349730838436 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_2}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\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}
t_0 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\
t_1 := -\frac{c}{b}\\
t_2 := \left(-b\right) - t_0\\
\mathbf{if}\;b \leq -3.96398836197757 \cdot 10^{+61}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{t_2}}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}\\

\mathbf{elif}\;b \leq 1.422349730838436 \cdot 10^{+93}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{t_2}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{c}{t_0 - b}\\


\end{array}\\

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

\mathbf{else}:\\
\;\;\;\;t_1\\


\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
 (let* ((t_0 (sqrt (- (* b b) (* (* a 4.0) c))))
        (t_1 (- (/ c b)))
        (t_2 (- (- b) t_0)))
   (if (<= b -3.96398836197757e+61)
     (if (>= b 0.0) (/ 1.0 (/ (* 2.0 a) t_2)) t_1)
     (if (<= b 1.422349730838436e+93)
       (if (>= b 0.0) (/ t_2 (* 2.0 a)) (* 2.0 (/ c (- t_0 b))))
       (if (>= b 0.0) (- (/ c b) (/ b a)) t_1)))))
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 t_0 = sqrt((b * b) - ((a * 4.0) * c));
	double t_1 = -(c / b);
	double t_2 = -b - t_0;
	double tmp_1;
	if (b <= -3.96398836197757e+61) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 1.0 / ((2.0 * a) / t_2);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.422349730838436e+93) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2 / (2.0 * a);
		} else {
			tmp_3 = 2.0 * (c / (t_0 - b));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

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 < -3.96398836197757e61

    1. Initial program 27.1

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

    2. Taylor expanded in b around -inf 3.6

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

    3. Applied clear-num_binary643.6

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

    if -3.96398836197757e61 < b < 1.422349730838436e93

    1. Initial program 8.5

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

    2. Applied *-un-lft-identity_binary648.5

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

    3. Applied times-frac_binary648.5

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

    if 1.422349730838436e93 < b

    1. Initial program 44.2

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

    2. Taylor expanded in b around -inf 44.2

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

    3. Taylor expanded in b around inf 4.2

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.96398836197757 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.422349730838436 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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)))))))