Average Error: 34.2 → 6.6
Time: 4.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -8.273540851308956 \cdot 10^{+145}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.561644019231982 \cdot 10^{-259}:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq 5.421378530068417 \cdot 10^{+117}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -8.273540851308956 \cdot 10^{+145}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq -3.561644019231982 \cdot 10^{-259}:\\
\;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right) \cdot \frac{0.5}{a}\\

\mathbf{elif}\;b \leq 5.421378530068417 \cdot 10^{+117}:\\
\;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b - c \cdot \left(a \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 -8.273540851308956e+145)
   (- (/ c b) (/ b a))
   (if (<= b -3.561644019231982e-259)
     (* (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (/ 0.5 a))
     (if (<= b 5.421378530068417e+117)
       (* c (/ -2.0 (+ b (sqrt (- (* b b) (* c (* a 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 <= -8.273540851308956e+145) {
		tmp = (c / b) - (b / a);
	} else if (b <= -3.561644019231982e-259) {
		tmp = (sqrt((b * b) - (c * (a * 4.0))) - b) * (0.5 / a);
	} else if (b <= 5.421378530068417e+117) {
		tmp = c * (-2.0 / (b + sqrt((b * b) - (c * (a * 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

Derivation

  1. Split input into 4 regimes

  2. if b < -8.27354085130895625e145

    1. Initial program 60.6

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

    2. Simplified60.6

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

    3. Taylor expanded around -inf 2.3

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

    if -8.27354085130895625e145 < b < -3.56164401923198198e-259

    1. Initial program 8.5

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

    2. Simplified8.5

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

    4. Applied div-inv_binary648.6

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

    5. Simplified8.6

      \[\leadsto \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \color{blue}{\frac{0.5}{a}}\]

    if -3.56164401923198198e-259 < b < 5.42137853006841731e117

    1. Initial program 31.6

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

    2. Simplified31.6

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

    4. Applied flip--_binary6431.7

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

    5. Simplified15.4

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

    6. Simplified15.4

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

    8. Applied *-un-lft-identity_binary6415.4

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

    9. Applied times-frac_binary6415.5

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

    10. Applied times-frac_binary6414.6

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

    11. Simplified8.8

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

    12. Simplified8.8

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

    if 5.42137853006841731e117 < b

    1. Initial program 60.8

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

    2. Simplified60.8

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

    3. Taylor expanded around inf 2.8

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

    4. Simplified2.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.273540851308956 \cdot 10^{+145}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.561644019231982 \cdot 10^{-259}:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;b \leq 5.421378530068417 \cdot 10^{+117}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))