quadm (p42, negative)

Average Error: 34.1 → 6.7

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.5617547656451965 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.6894731563618579 \cdot 10^{-274}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2} \cdot \frac{4 \cdot c}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1.8284348008013831 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

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 VAR;
	if ((b <= -4.5617547656451965e+153)) {
		VAR = (-1.0 * (c / b));
	} else {
		double VAR_1;
		if ((b <= -1.6894731563618579e-274)) {
			VAR_1 = (((1.0 / sqrt((sqrt(((b * b) - (4.0 * (a * c)))) - b))) / 2.0) * ((4.0 * c) / sqrt((sqrt(((b * b) - (4.0 * (a * c)))) - b))));
		} else {
			double VAR_2;
			if ((b <= 1.828434800801383e+151)) {
				VAR_2 = (1.0 / ((2.0 * a) / (-b - sqrt(((b * b) - (4.0 * (a * c)))))));
			} else {
				VAR_2 = (1.0 * ((c / b) - (b / a)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.1
Target	20.9
Herbie	6.7

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -4.5617547656451965e+153

   1. Initial program 63.9

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

   2. Taylor expanded around -inf 1.5

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

   if -4.5617547656451965e+153 < b < -1.6894731563618579e-274

   1. Initial program 36.2

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

   3. Applied flip-- 36.2

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

   4. Simplified 15.6

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

   5. Simplified 15.6

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

   7. Applied add-sqr-sqrt 15.7

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

   8. Applied *-un-lft-identity 15.7

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

   9. Applied times-frac 15.7

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

   10. Applied times-frac 14.9

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

   11. Simplified 14.4

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

   13. Applied *-un-lft-identity 14.4

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

   14. Applied times-frac 14.4

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

   15. Simplified 14.4

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

   16. Simplified 8.0

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

   if -1.6894731563618579e-274 < b < 1.828434800801383e+151

   1. Initial program 9.3

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

   3. Applied clear-num 9.5

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

   if 1.828434800801383e+151 < b

   1. Initial program 62.9

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

   2. Taylor expanded around inf 2.1

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

   3. Simplified 2.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.5617547656451965 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.6894731563618579 \cdot 10^{-274}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2} \cdot \frac{4 \cdot c}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1.8284348008013831 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))