Average Error: 33.8 → 9.8
Time: 8.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.941647724720173 \cdot 10^{+151}:\\ \;\;\;\;\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1356574979833072 \cdot 10^{-83}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -1.941647724720173 \cdot 10^{+151}:\\
\;\;\;\;\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right) \cdot 0.5\\

\mathbf{elif}\;b \leq 3.1356574979833072 \cdot 10^{-83}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a}\\

\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 -1.941647724720173e+151)
   (* (- (* 2.0 (/ c b)) (* 2.0 (/ b a))) 0.5)
   (if (<= b 3.1356574979833072e-83)
     (* 0.5 (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) a))
     (- (/ 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 <= -1.941647724720173e+151) {
		tmp = ((2.0 * (c / b)) - (2.0 * (b / a))) * 0.5;
	} else if (b <= 3.1356574979833072e-83) {
		tmp = 0.5 * ((sqrt((b * b) - (4.0 * (c * a))) - b) / a);
	} 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 < -1.94164772472017305e151

    1. Initial program 62.7

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

    2. Simplified62.7

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

    4. Applied clear-num_binary6462.7

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

    5. Simplified62.7

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

    7. Applied associate-/r/_binary6462.7

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

    8. Applied add-sqr-sqrt_binary6462.7

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

    9. Applied times-frac_binary6462.7

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

    10. Simplified62.7

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

    11. Simplified62.7

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

    12. Taylor expanded around -inf 2.1

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)} \cdot 0.5\]

    if -1.94164772472017305e151 < b < 3.13565749798330719e-83

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied clear-num_binary6412.1

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

    5. Simplified12.1

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

    7. Applied associate-/r/_binary6412.1

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

    8. Applied add-sqr-sqrt_binary6412.1

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

    9. Applied times-frac_binary6412.1

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

    10. Simplified12.0

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

    11. Simplified12.0

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

    if 3.13565749798330719e-83 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 9.2

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

    4. Simplified9.2

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.941647724720173 \cdot 10^{+151}:\\ \;\;\;\;\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1356574979833072 \cdot 10^{-83}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021113 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))