Average Error: 19.8 → 6.5
Time: 7.7s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
↓
\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_2 := \frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{if}\;b \leq -2.304620787713193 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{t_1 - b}{2 \cdot a}\\ \mathbf{if}\;b \leq 1.5009182714254429 \cdot 10^{+118}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\\ \end{array} \]
\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}
t_0 := \left(-b\right) - b\\
t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\
t_2 := \frac{2 \cdot c}{\left(-b\right) - t_1}\\
\mathbf{if}\;b \leq -2.304620787713193 \cdot 10^{+101}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;t_2\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0}{2 \cdot a}\\


\end{array}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_3 := \frac{t_1 - b}{2 \cdot a}\\
\mathbf{if}\;b \leq 1.5009182714254429 \cdot 10^{+118}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;t_2\\

\mathbf{else}:\\
\;\;\;\;t_3\\


\end{array}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot c}{t_0}\\

\mathbf{else}:\\
\;\;\;\;t_3\\


\end{array}\\


\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
 (let* ((t_0 (- (- b) b))
        (t_1 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_2 (/ (* 2.0 c) (- (- b) t_1))))
   (if (<= b -2.304620787713193e+101)
     (if (>= b 0.0) t_2 (/ t_0 (* 2.0 a)))
     (let* ((t_3 (/ (- t_1 b) (* 2.0 a))))
       (if (<= b 1.5009182714254429e+118)
         (if (>= b 0.0) t_2 t_3)
         (if (>= b 0.0) (/ (* 2.0 c) t_0) t_3))))))
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 t_0 = -b - b;
	double t_1 = sqrt((b * b) - (c * (4.0 * a)));
	double t_2 = (2.0 * c) / (-b - t_1);
	double tmp_1;
	if (b <= -2.304620787713193e+101) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else {
		double t_3 = (t_1 - b) / (2.0 * a);
		double tmp_4;
		if (b <= 1.5009182714254429e+118) {
			double tmp_5;
			if (b >= 0.0) {
				tmp_5 = t_2;
			} else {
				tmp_5 = t_3;
			}
			tmp_4 = tmp_5;
		} else if (b >= 0.0) {
			tmp_4 = (2.0 * c) / t_0;
		} else {
			tmp_4 = t_3;
		}
		tmp_1 = tmp_4;
	}
	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 < -2.30462078771319297e101

    1. Initial program 47.9

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

    2. Taylor expanded in b around -inf 3.9

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

    if -2.30462078771319297e101 < b < 1.5009182714254429e118

    1. Initial program 8.6

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

    if 1.5009182714254429e118 < b

    1. Initial program 31.7

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

    2. Taylor expanded in b around inf 2.2

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.304620787713193 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5009182714254429 \cdot 10^{+118}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))