Average Error: 34.5 → 10.3
Time: 6.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -4.266218223523121 \cdot 10^{+118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.399917751723213 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.266218223523121 \cdot 10^{+118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \leq 8.399917751723213 \cdot 10^{-42}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \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 -4.266218223523121e+118)
   (* 1.0 (- (/ c b) (/ b a)))
   (if (<= b 8.399917751723213e-42)
     (/ 1.0 (/ a (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) 2.0)))
     (- (* 1.0 (/ 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 <= -4.266218223523121e+118)) {
		tmp = ((double) (1.0 * ((double) ((c / b) - (b / a)))));
	} else {
		double tmp_1;
		if ((b <= 8.399917751723213e-42)) {
			tmp_1 = (1.0 / (a / (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (c * a)))))))) - b)) / 2.0)));
		} else {
			tmp_1 = ((double) -(((double) (1.0 * (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

Original34.5
Target21.4
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.2662182235231212e118

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around -inf 3.3

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

    4. Simplified3.3

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

    if -4.2662182235231212e118 < b < 8.3999177517232133e-42

    1. Initial program 14.5

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

    2. Simplified14.5

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

    4. Applied clear-num_binary6414.6

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

    5. Simplified14.6

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

    if 8.3999177517232133e-42 < b

    1. Initial program 54.9

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

    2. Simplified54.9

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

    3. Taylor expanded around inf 7.1

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.266218223523121 \cdot 10^{+118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 8.399917751723213 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))