Average Error: 33.7 → 8.2
Time: 4.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3667361231028956 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.59554765878054 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.1883717871229594 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{a}{\sqrt{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \cdot \frac{c \cdot -4}{\sqrt{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -1.3667361231028956 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq -7.59554765878054 \cdot 10^{-309}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 1.1883717871229594 \cdot 10^{+34}:\\
\;\;\;\;\frac{\frac{a}{\sqrt{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \cdot \frac{c \cdot -4}{\sqrt{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}{a \cdot 2}\\

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

\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 -1.3667361231028956e+152)
   (- (/ c b) (/ b a))
   (if (<= b -7.59554765878054e-309)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (if (<= b 1.1883717871229594e+34)
       (/
        (*
         (/ a (sqrt (+ b (sqrt (+ (* b b) (* a (* c -4.0)))))))
         (/ (* c -4.0) (sqrt (+ b (sqrt (+ (* b b) (* a (* c -4.0))))))))
        (* a 2.0))
       (- (/ c b))))))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= -1.3667361231028956e+152)) {
		tmp = ((double) ((c / b) - (b / a)));
	} else {
		double tmp_1;
		if ((b <= -7.59554765878054e-309)) {
			tmp_1 = (((double) (((double) sqrt(((double) (((double) (b * b)) + ((double) (a * ((double) (c * -4.0)))))))) - b)) / ((double) (a * 2.0)));
		} else {
			double tmp_2;
			if ((b <= 1.1883717871229594e+34)) {
				tmp_2 = (((double) ((a / ((double) sqrt(((double) (b + ((double) sqrt(((double) (((double) (b * b)) + ((double) (a * ((double) (c * -4.0))))))))))))) * (((double) (c * -4.0)) / ((double) sqrt(((double) (b + ((double) sqrt(((double) (((double) (b * b)) + ((double) (a * ((double) (c * -4.0))))))))))))))) / ((double) (a * 2.0)));
			} else {
				tmp_2 = ((double) -((c / b)));
			}
			tmp_1 = tmp_2;
		}
		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
Target21.1
Herbie8.2
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.36673612310289565e152

    1. Initial program 63.4

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

    2. Simplified63.4

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

    3. Taylor expanded around -inf 2.3

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

    if -1.36673612310289565e152 < b < -7.5955476587805401e-309

    1. Initial program 8.8

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

    2. Simplified8.8

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

    4. Applied sub-neg_binary648.8

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

    5. Simplified8.8

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

    if -7.5955476587805401e-309 < b < 1.1883717871229594e34

    1. Initial program 27.8

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

    2. Simplified27.8

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

    4. Applied sub-neg_binary6427.8

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

    5. Simplified27.9

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

    7. Applied flip--_binary6427.9

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

    8. Simplified17.4

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

    9. Simplified17.4

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

    11. Applied add-sqr-sqrt_binary6417.6

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

    12. Applied times-frac_binary6414.9

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

    if 1.1883717871229594e34 < b

    1. Initial program 56.6

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

    2. Simplified56.6

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

    3. Taylor expanded around inf 4.6

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

    4. Simplified4.6

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3667361231028956 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.59554765878054 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.1883717871229594 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{a}{\sqrt{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}} \cdot \frac{c \cdot -4}{\sqrt{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected #f

  :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)))