quadp (p42, positive)

Average Error: 34.2 → 7.0

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.7863758169125638 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.96206905407944466 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot 2}\\ \mathbf{elif}\;b \le 1.7024537441103181 \cdot 10^{28}:\\ \;\;\;\;\frac{4 \cdot c}{\left(2 \cdot a\right) \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot c}{-4 \cdot b}\\ \end{array}\]

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 <= -2.7863758169125638e+138)) {
		VAR = (1.0 * ((c / b) - (b / a)));
	} else {
		double VAR_1;
		if ((b <= -4.962069054079445e-170)) {
			VAR_1 = (1.0 / ((a / (sqrt(((b * b) - (4.0 * (a * c)))) - b)) * 2.0));
		} else {
			double VAR_2;
			if ((b <= 1.702453744110318e+28)) {
				VAR_2 = ((4.0 * c) / ((2.0 * a) * ((-b - sqrt(((b * b) - (4.0 * (a * c))))) / a)));
			} else {
				VAR_2 = ((4.0 * c) / (-4.0 * b));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	20.9
Herbie	7.0

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

Derivation

1. Split input into 4 regimes

2. if b < -2.7863758169125638e+138

   1. Initial program 58.6

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

   2. Taylor expanded around -inf 2.6

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

   3. Simplified 2.6

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

   if -2.7863758169125638e+138 < b < -4.962069054079445e-170

   1. Initial program 5.2

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

   3. Applied clear-num 5.4

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

   4. Simplified 5.4

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

   if -4.962069054079445e-170 < b < 1.702453744110318e+28

   1. Initial program 25.3

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

   3. Applied flip-+ 25.6

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

   4. Simplified 17.8

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

   6. Applied *-commutative 17.8

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

   7. Applied associate-*r* 17.9

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

   8. Applied associate-/l* 15.6

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

   9. Applied associate-/l/ 12.0

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

   if 1.702453744110318e+28 < b

   1. Initial program 57.0

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

   3. Applied flip-+ 57.0

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

   4. Simplified 28.1

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

   6. Applied *-commutative 28.1

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

   7. Applied associate-*r* 28.2

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

   8. Applied associate-/l* 29.9

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

   9. Applied associate-/l/ 28.1

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

   10. Taylor expanded around 0 4.9

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

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7863758169125638 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.96206905407944466 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot 2}\\ \mathbf{elif}\;b \le 1.7024537441103181 \cdot 10^{28}:\\ \;\;\;\;\frac{4 \cdot c}{\left(2 \cdot a\right) \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot c}{-4 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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