quadm (p42, negative)

Average Error: 34.2 → 6.6

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.4767676629755074 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

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 temp;
	if ((b <= -4.4767676629755074e+150)) {
		temp = (-1.0 * (c / b));
	} else {
		double temp_1;
		if ((b <= -9.7864508816149e-248)) {
			temp_1 = ((1.0 / 2.0) * ((4.0 * c) / (sqrt(((b * b) - (4.0 * (a * c)))) - b)));
		} else {
			double temp_2;
			if ((b <= 8.293479107059485e+98)) {
				temp_2 = ((-b / (2.0 * a)) - (sqrt(((b * b) - (4.0 * (a * c)))) / (2.0 * a)));
			} else {
				temp_2 = (1.0 * ((c / b) - (b / a)));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.5
Herbie	6.6

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

Derivation

1. Split input into 4 regimes

2. if b < -4.4767676629755074e+150

   1. Initial program 63.7

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

   2. Taylor expanded around -inf 1.4

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

   if -4.4767676629755074e+150 < b < -9.7864508816149e-248

   1. Initial program 36.5

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

   3. Applied flip-- 36.5

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

   4. Simplified 16.2

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

   5. Simplified 16.2

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

   7. Applied *-un-lft-identity 16.2

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

   8. Applied *-un-lft-identity 16.2

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

   9. Applied times-frac 16.2

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

   10. Applied times-frac 16.2

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

   11. Simplified 16.2

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

   12. Simplified 14.2

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

   14. Applied *-un-lft-identity 14.2

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

   15. Applied times-frac 14.2

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

   16. Simplified 14.2

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

   17. Simplified 7.3

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

   if -9.7864508816149e-248 < b < 8.293479107059485e+98

   1. Initial program 10.0

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

   3. Applied div-sub 10.0

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

   if 8.293479107059485e+98 < b

   1. Initial program 46.8

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

   2. Taylor expanded around inf 3.8

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

   3. Simplified 3.8

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.4767676629755074 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))