Average Error: 34.1 → 9.1
Time: 12.9s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.0821358365922274 \cdot 10^{+128}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;b \leq 1.30094701045847 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.640574928265682 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{t_0}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \end{array} \]
\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 -1.0821358365922274 \cdot 10^{+128}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := c \cdot \left(a \cdot -4\right)\\
\mathbf{if}\;b \leq 1.30094701045847 \cdot 10^{-121}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 6.640574928265682 \cdot 10^{+57}:\\
\;\;\;\;\frac{\frac{t_0}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a \cdot 2}\\

\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 -1.0821358365922274e+128)
   (- (/ c b) (/ b a))
   (let* ((t_0 (* c (* a -4.0))))
     (if (<= b 1.30094701045847e-121)
       (/ (- (sqrt (+ (* b b) t_0)) b) (* a 2.0))
       (if (<= b 6.640574928265682e+57)
         (/ (/ t_0 (+ b (sqrt (- (* b b) (* 4.0 (* c a)))))) (* a 2.0))
         (- (/ 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 <= -1.0821358365922274e+128) {
		tmp = (c / b) - (b / a);
	} else {
		double t_0 = c * (a * -4.0);
		double tmp_1;
		if (b <= 1.30094701045847e-121) {
			tmp_1 = (sqrt((b * b) + t_0) - b) / (a * 2.0);
		} else if (b <= 6.640574928265682e+57) {
			tmp_1 = (t_0 / (b + sqrt((b * b) - (4.0 * (c * a))))) / (a * 2.0);
		} else {
			tmp_1 = -(c / b);
		}
		tmp = tmp_1;
	}
	return tmp;
}

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

Target

Original34.1
Target21.0
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 < -1.08213583659222744e128

    1. Initial program 53.9

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

    2. Simplified53.9

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

    3. Taylor expanded around -inf 2.9

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

    if -1.08213583659222744e128 < b < 1.3009470104584699e-121

    1. Initial program 12.0

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

    2. Simplified12.0

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}} \]
    3. Using strategy rm

    4. Applied sub-neg_binary6412.0

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

    5. Simplified12.0

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

    if 1.3009470104584699e-121 < b < 6.6405749282656817e57

    1. Initial program 39.5

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

    2. Simplified39.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}} \]
    3. Using strategy rm

    4. Applied sub-neg_binary6439.5

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

    5. Simplified39.6

      \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}} - b}{a \cdot 2} \]
    6. Using strategy rm

    7. Applied flip--_binary6439.6

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

    8. Simplified16.7

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot -4\right) + 0}}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} + b}}{a \cdot 2} \]

    9. Simplified16.7

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot -4\right) + 0}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{a \cdot 2} \]

    if 6.6405749282656817e57 < b

    1. Initial program 57.5

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

    2. Simplified57.5

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

    3. Taylor expanded around inf 3.4

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

    4. Simplified3.4

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.0821358365922274 \cdot 10^{+128}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.30094701045847 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.640574928265682 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot -4\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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)))