Quadratic roots, full range

Average Error: 34.5 → 6.8

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3014879836557515 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.8145284637786566 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 6.972667102848289 \cdot 10^{+99}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

FPCore

(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 -1.3014879836557515e+97)
   (- (/ c b) (/ b a))
   (if (<= b -1.8145284637786566e-293)
     (/ 1.0 (/ a (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) 2.0)))
     (if (<= b 6.972667102848289e+99)
       (/ (* c -2.0) (+ b (sqrt (- (* b b) (* c (* a 4.0))))))
       (- (/ c b))))))

C

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 <= -1.3014879836557515e+97) {
		tmp = (c / b) - (b / a);
	} else if (b <= -1.8145284637786566e-293) {
		tmp = 1.0 / (a / ((sqrt((b * b) - (c * (a * 4.0))) - b) / 2.0));
	} else if (b <= 6.972667102848289e+99) {
		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

Derivation

1. Split input into 4 regimes

2. if b < -1.3014879836557515e97

   1. Initial program 46.0

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

   2. Simplified 46.0

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

   3. Taylor expanded around -inf 3.9

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

   if -1.3014879836557515e97 < b < -1.81452846377865664e-293

   1. Initial program 9.6

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

   2. Simplified 9.6

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

   4. Applied clear-num_binary64_422 9.7

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

   5. Simplified 9.7

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

   if -1.81452846377865664e-293 < b < 6.97266710284828931e99

   1. Initial program 31.6

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

   2. Simplified 31.6

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

   4. Applied flip--_binary64_398 31.7

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

   5. Simplified 16.2

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

   6. Simplified 16.2

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

   8. Applied div-inv_binary64_420 16.2

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

   9. Applied times-frac_binary64_429 15.2

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

   10. Simplified 9.0

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

   11. Simplified 9.0

       \[\leadsto \left(c \cdot -4\right) \cdot \color{blue}{\frac{0.5}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]
   12. Using strategy rm

   13. Applied associate-*r/_binary64_365 8.8

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

   14. Simplified 8.8

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

   if 6.97266710284828931e99 < b

   1. Initial program 59.4

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

   2. Simplified 59.4

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

   3. Taylor expanded around inf 2.7

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

   4. Simplified 2.7

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3014879836557515 \cdot 10^{+97}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.8145284637786566 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 6.972667102848289 \cdot 10^{+99}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))