Average Error: 34.3 → 6.7
Time: 6.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3138164296877506 \cdot 10^{+98}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.621813580046727 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 7.7377607878944895 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{c \cdot -4}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -1.3138164296877506 \cdot 10^{+98}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq -7.621813580046727 \cdot 10^{-260}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 7.7377607878944895 \cdot 10^{+96}:\\
\;\;\;\;\frac{1}{\frac{2}{\frac{c \cdot -4}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\

\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 -1.3138164296877506e+98)
   (- (/ c b) (/ b a))
   (if (<= b -7.621813580046727e-260)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (if (<= b 7.7377607878944895e+96)
       (/ 1.0 (/ 2.0 (/ (* c -4.0) (+ b (sqrt (+ (* b b) (* a (* c -4.0))))))))
       (- (/ 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 <= -1.3138164296877506e+98) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7.621813580046727e-260) {
		tmp = (sqrt((b * b) + (a * (c * -4.0))) - b) / (a * 2.0);
	} else if (b <= 7.7377607878944895e+96) {
		tmp = 1.0 / (2.0 / ((c * -4.0) / (b + sqrt((b * b) + (a * (c * -4.0))))));
	} 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

Original34.3
Target21.0
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.31381642968775059e98

    1. Initial program 47.4

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

    2. Simplified47.4

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

    3. Taylor expanded around -inf 3.3

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

    if -1.31381642968775059e98 < b < -7.62181358004672682e-260

    1. Initial program 8.3

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

    2. Simplified8.3

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

    4. Applied sub-neg_binary64_4078.3

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

    5. Simplified8.3

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

    if -7.62181358004672682e-260 < b < 7.7377607878944895e96

    1. Initial program 30.9

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

    2. Simplified30.9

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

    4. Applied sub-neg_binary64_40730.9

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

    5. Simplified31.0

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

    7. Applied flip--_binary64_38931.0

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

    8. Simplified16.7

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

    9. Simplified16.7

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

    11. Applied clear-num_binary64_41316.9

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

    12. Simplified10.1

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

    if 7.7377607878944895e96 < b

    1. Initial program 59.5

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

    2. Simplified59.5

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

    3. Taylor expanded around inf 2.6

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

    4. Simplified2.6

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3138164296877506 \cdot 10^{+98}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.621813580046727 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 7.7377607878944895 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{c \cdot -4}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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)))