Average Error: 33.8 → 9.1
Time: 10.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.747572511563342 \cdot 10^{+134}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq 3.9562943804830147 \cdot 10^{-153}:\\ \;\;\;\;\frac{\left(t_0 - b\right) \cdot 0.5}{a}\\ \mathbf{elif}\;b \leq 2.3361535024269667 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \frac{0.5}{a}}{b + t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -1.747572511563342 \cdot 10^{+134}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\
\mathbf{if}\;b \leq 3.9562943804830147 \cdot 10^{-153}:\\
\;\;\;\;\frac{\left(t_0 - b\right) \cdot 0.5}{a}\\

\mathbf{elif}\;b \leq 2.3361535024269667 \cdot 10^{+101}:\\
\;\;\;\;\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \frac{0.5}{a}}{b + t_0}\\

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


\end{array}\\


\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -1.747572511563342e+134)
   (- (/ c b) (/ b a))
   (let* ((t_0 (sqrt (fma a (* c -4.0) (* b b)))))
     (if (<= b 3.9562943804830147e-153)
       (/ (* (- t_0 b) 0.5) a)
       (if (<= b 2.3361535024269667e+101)
         (/ (* (* c (* a -4.0)) (/ 0.5 a)) (+ b t_0))
         (- (/ c b)))))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - (4.0 * (a * c)))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.747572511563342e+134) {
		tmp = (c / b) - (b / a);
	} else {
		double t_0 = sqrt(fma(a, (c * -4.0), (b * b)));
		double tmp_1;
		if (b <= 3.9562943804830147e-153) {
			tmp_1 = ((t_0 - b) * 0.5) / a;
		} else if (b <= 2.3361535024269667e+101) {
			tmp_1 = ((c * (a * -4.0)) * (0.5 / a)) / (b + t_0);
		} else {
			tmp_1 = -(c / b);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.8
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.74757251156334196e134

    1. Initial program 55.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Taylor expanded in b around -inf 3.1

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

    if -1.74757251156334196e134 < b < 3.9562943804830147e-153

    1. Initial program 11.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified11.2

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

    3. Applied associate-*r/_binary6411.1

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}} \]

    if 3.9562943804830147e-153 < b < 2.3361535024269667e101

    1. Initial program 39.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified39.2

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

    3. Applied flip--_binary6439.2

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}} \cdot \frac{0.5}{a} \]

    4. Applied associate-*l/_binary6439.3

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b}} \]

    5. Simplified16.0

      \[\leadsto \frac{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + 0\right) \cdot \frac{0.5}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + b} \]

    if 2.3361535024269667e101 < b

    1. Initial program 59.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Taylor expanded in b around inf 2.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

    3. Simplified2.5

      \[\leadsto \color{blue}{-\frac{c}{b}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.747572511563342 \cdot 10^{+134}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9562943804830147 \cdot 10^{-153}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{elif}\;b \leq 2.3361535024269667 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(c \cdot \left(a \cdot -4\right)\right) \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))