Average Error: 34.5 → 9.5
Time: 7.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -4.270668456863539 \cdot 10^{+83}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.360570160473105 \cdot 10^{-229}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 8.921968209640008 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -4.270668456863539 \cdot 10^{+83}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 3.360570160473105 \cdot 10^{-229}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}}\\

\mathbf{elif}\;b \leq 8.921968209640008 \cdot 10^{+83}:\\
\;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{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 -4.270668456863539e+83)
   (- (/ c b) (/ b a))
   (if (<= b 3.360570160473105e-229)
     (/ 1.0 (/ a (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) 2.0)))
     (if (<= b 8.921968209640008e+83)
       (/
        (/ (* a (* c -4.0)) (+ b (sqrt (+ (* b b) (* a (* c -4.0))))))
        (* 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 <= -4.270668456863539e+83) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.360570160473105e-229) {
		tmp = 1.0 / (a / ((sqrt((b * b) - (4.0 * (c * a))) - b) / 2.0));
	} else if (b <= 8.921968209640008e+83) {
		tmp = ((a * (c * -4.0)) / (b + sqrt((b * b) + (a * (c * -4.0))))) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	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
Target21.5
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.27066845686353916e83

    1. Initial program 43.5

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

    2. Simplified43.5

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

    3. Taylor expanded around -inf 4.8

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

    if -4.27066845686353916e83 < b < 3.36057016047310531e-229

    1. Initial program 11.0

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

    2. Simplified11.0

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

    4. Applied clear-num_binary6411.1

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

    5. Simplified11.1

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

    if 3.36057016047310531e-229 < b < 8.9219682096400083e83

    1. Initial program 35.6

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

    2. Simplified35.6

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

    4. Applied sub-neg_binary6435.6

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

    5. Simplified35.6

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

    7. Applied flip--_binary6435.7

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

    8. Simplified16.8

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

    9. Simplified16.8

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

    if 8.9219682096400083e83 < b

    1. Initial program 58.3

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

    2. Simplified58.3

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

    3. Taylor expanded around inf 3.1

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

    4. Simplified3.1

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.270668456863539 \cdot 10^{+83}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.360570160473105 \cdot 10^{-229}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 8.921968209640008 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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)))