Average Error: 19.4 → 6.5
Time: 10.6s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -9.602324171183254 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{c \cdot 2}{t_0 - b}\\ \mathbf{if}\;b \leq 1.8712767159114232 \cdot 10^{+66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, t_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \end{array} \]
\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{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\
\mathbf{if}\;b \leq -9.602324171183254 \cdot 10^{+93}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\

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


\end{array}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{c \cdot 2}{t_0 - b}\\
\mathbf{if}\;b \leq 1.8712767159114232 \cdot 10^{+66}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, t_0\right)}{a}\\

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


\end{array}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;-0.5 \cdot \frac{b + b}{a}\\

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


\end{array}\\


\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 (fma a (* c -4.0) (* b b)))))
   (if (<= b -9.602324171183254e+93)
     (if (>= b 0.0) (* -0.5 (/ (+ b t_0) a)) (/ (* c 2.0) (- (- b) b)))
     (let* ((t_1 (/ (* c 2.0) (- t_0 b))))
       (if (<= b 1.8712767159114232e+66)
         (if (>= b 0.0) (* -0.5 (/ (fma (sqrt b) (sqrt b) t_0) a)) t_1)
         (if (>= b 0.0) (* -0.5 (/ (+ 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(fma(a, (c * -4.0), (b * b)));
	double tmp_1;
	if (b <= -9.602324171183254e+93) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else {
		double t_1 = (c * 2.0) / (t_0 - b);
		double tmp_4;
		if (b <= 1.8712767159114232e+66) {
			double tmp_5;
			if (b >= 0.0) {
				tmp_5 = -0.5 * (fma(sqrt(b), sqrt(b), t_0) / a);
			} else {
				tmp_5 = t_1;
			}
			tmp_4 = tmp_5;
		} else if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + b) / a);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	}
	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 < -9.60232417118325409e93

    1. Initial program 28.8

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

    2. Simplified28.7

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

    3. Taylor expanded in b around -inf 2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]

    if -9.60232417118325409e93 < b < 1.87127671591142321e66

    1. Initial program 8.6

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

    2. Simplified8.6

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

    3. Applied add-sqr-sqrt_binary648.6

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

    4. Applied fma-def_binary648.6

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

    if 1.87127671591142321e66 < b

    1. Initial program 40.4

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

    2. Simplified40.5

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

    3. Taylor expanded in a around 0 5.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.602324171183254 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.8712767159114232 \cdot 10^{+66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))))))