Average Error: 33.7 → 6.8
Time: 6.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.061643530149749 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -9.859859419540874 \cdot 10^{-169}:\\ \;\;\;\;\frac{t_0}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.9972411182687842 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c \cdot -2}{b + t_0}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \end{array} \]
\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 -3.061643530149749 \cdot 10^{+106}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\\
\mathbf{if}\;b \leq -9.859859419540874 \cdot 10^{-169}:\\
\;\;\;\;\frac{t_0}{a \cdot 2} - \frac{b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 1.9972411182687842 \cdot 10^{+77}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{c \cdot -2}{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 -3.061643530149749e+106)
   (- (/ c b) (/ b a))
   (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* c a))))))
     (if (<= b -9.859859419540874e-169)
       (- (/ t_0 (* a 2.0)) (/ b (* a 2.0)))
       (if (<= b 1.9972411182687842e+77)
         (/ 1.0 (/ 1.0 (/ (* c -2.0) (+ 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 <= -3.061643530149749e+106) {
		tmp = (c / b) - (b / a);
	} else {
		double t_0 = sqrt((b * b) - (4.0 * (c * a)));
		double tmp_1;
		if (b <= -9.859859419540874e-169) {
			tmp_1 = (t_0 / (a * 2.0)) - (b / (a * 2.0));
		} else if (b <= 1.9972411182687842e+77) {
			tmp_1 = 1.0 / (1.0 / ((c * -2.0) / (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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.7
Target20.6
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.061643530149749e106

    1. Initial program 48.7

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

    2. Simplified48.7

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

    3. Taylor expanded around -inf 3.3

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

    if -3.061643530149749e106 < b < -9.85985941954087442e-169

    1. Initial program 6.0

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

    2. Simplified6.0

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

    4. Applied div-sub_binary646.0

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

    if -9.85985941954087442e-169 < b < 1.99724111826878415e77

    1. Initial program 27.9

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

    2. Simplified27.9

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

    4. Applied sub-neg_binary6427.9

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

    5. Simplified27.9

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

    7. Applied flip--_binary6428.1

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

    8. Simplified16.9

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

    9. Simplified16.9

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

    11. Applied clear-num_binary6417.0

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\frac{c \cdot \left(a \cdot -4\right) + 0}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}} \]

    12. Simplified17.0

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{-2 \cdot \frac{c \cdot a}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}} \]
    13. Using strategy rm

    14. Applied clear-num_binary6417.0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{-2 \cdot \frac{c \cdot a}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}}} \]

    15. Simplified10.9

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{c \cdot -2}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}} \]

    if 1.99724111826878415e77 < 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 simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.061643530149749 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -9.859859419540874 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.9972411182687842 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c \cdot -2}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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)))