Average Error: 34.0 → 10.4
Time: 12.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -4.029250158308429 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7994370256410445 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\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 -4.029250158308429 \cdot 10^{+153}:\\
\;\;\;\;-\frac{b}{a}\\

\mathbf{elif}\;b \leq 1.7994370256410445 \cdot 10^{-25}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{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 -4.029250158308429e+153)
   (- (/ b a))
   (if (<= b 1.7994370256410445e-25)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 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 <= -4.029250158308429e+153) {
		tmp = -(b / a);
	} else if (b <= 1.7994370256410445e-25) {
		tmp = (sqrt((b * b) - ((a * 4.0) * c)) - b) / (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

Derivation

  1. Split input into 3 regimes

  2. if b < -4.02925015830842898e153

    1. Initial program 63.9

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

    2. Applied add-exp-log_binary6463.9

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

    3. Simplified38.6

      \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b\right)}}}{2 \cdot a} \]

    4. Taylor expanded in b around -inf 2.6

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

    5. Simplified2.6

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

    if -4.02925015830842898e153 < b < 1.79943702564104447e-25

    1. Initial program 14.3

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

    if 1.79943702564104447e-25 < b

    1. Initial program 54.7

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

    2. Taylor expanded in b around inf 6.7

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

    3. Simplified6.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.029250158308429 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7994370256410445 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))