Average Error: 19.7 → 8.3
Time: 8.1s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
↓
\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ \mathbf{if}\;b \leq -4.439565493803235 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.029795161598339 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array} \]
\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}
t_0 := \left(-b\right) - b\\
t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\
\mathbf{if}\;b \leq -4.439565493803235 \cdot 10^{-48}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\

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


\end{array}\\

\mathbf{elif}\;b \leq 5.029795161598339 \cdot 10^{+117}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5 \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)}{a}\\


\end{array}\\

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

\mathbf{else}:\\
\;\;\;\;\frac{t_1 - 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
 (let* ((t_0 (- (- b) b)) (t_1 (sqrt (- (* b b) (* c (* 4.0 a))))))
   (if (<= b -4.439565493803235e-48)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) (/ t_0 (* 2.0 a)))
     (if (<= b 5.029795161598339e+117)
       (if (>= b 0.0)
         (* c (/ -2.0 (+ b (sqrt (fma a (* c -4.0) (* b b))))))
         (/ (* 0.5 (- (hypot b (sqrt (* a (* c -4.0)))) b)) a))
       (if (>= b 0.0) (/ (* 2.0 c) t_0) (/ (- t_1 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 t_0 = -b - b;
	double t_1 = sqrt((b * b) - (c * (4.0 * a)));
	double tmp_1;
	if (b <= -4.439565493803235e-48) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_2 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.029795161598339e+117) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + sqrt(fma(a, (c * -4.0), (b * b)))));
		} else {
			tmp_3 = (0.5 * (hypot(b, sqrt(a * (c * -4.0))) - b)) / a;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / t_0;
	} else {
		tmp_1 = (t_1 - b) / (2.0 * a);
	}
	return tmp_1;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -4.4395654938032352e-48

    1. Initial program 28.7

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

    2. Taylor expanded in b around -inf 9.9

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

    if -4.4395654938032352e-48 < b < 5.0297951615983392e117

    1. Initial program 9.7

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

    2. Simplified9.8

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

    3. Applied associate-*r/_binary649.7

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

    4. Simplified10.2

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

    if 5.0297951615983392e117 < b

    1. Initial program 31.0

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

    2. Taylor expanded in b around inf 1.9

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.439565493803235 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.029795161598339 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) - b\right)}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(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))))