quad2p (problem 3.2.1, positive)

Average Error: 33.9 → 7.4

Time: 5.9s

Precision: 64

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.86870705061354704 \cdot 10^{41}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.4521535344124743 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.8543159123016849 \cdot 10^{89}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double code(double a, double b_2, double c) {
	return ((double) (((double) (((double) -(b_2)) + ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (a * c)))))))) / a));
}

↓

double code(double a, double b_2, double c) {
	double VAR;
	if ((b_2 <= -3.868707050613547e+41)) {
		VAR = ((double) (((double) (0.5 * ((double) (c / b_2)))) - ((double) (2.0 * ((double) (b_2 / a))))));
	} else {
		double VAR_1;
		if ((b_2 <= -1.4521535344124743e-304)) {
			VAR_1 = ((double) (((double) (((double) -(b_2)) + ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (a * c)))))))) * ((double) (1.0 / a))));
		} else {
			double VAR_2;
			if ((b_2 <= 1.854315912301685e+89)) {
				VAR_2 = ((double) (((double) (1.0 / ((double) (((double) -(b_2)) - ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (a * c)))))))))) * ((double) (((double) (((double) cbrt(a)) / ((double) (1.0 / c)))) / ((double) cbrt(a))))));
			} else {
				VAR_2 = ((double) (-0.5 * ((double) (c / b_2))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.868707050613547e+41

   1. Initial program 37.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 4.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   if -3.868707050613547e+41 < b_2 < -1.4521535344124743e-304

   1. Initial program 9.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied div-inv 10.0

      \[\leadsto \color{blue}{\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

   if -1.4521535344124743e-304 < b_2 < 1.854315912301685e+89

   1. Initial program 31.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-+ 31.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 16.0

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

   7. Applied associate-/r* 16.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{1}}{a}}\]

   8. Simplified 14.0

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}{a}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 14.7

       \[\leadsto \frac{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   11. Applied div-inv 14.7

       \[\leadsto \frac{\frac{a}{\color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   12. Applied add-cube-cbrt 14.0

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{c}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   13. Applied times-frac 14.3

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\sqrt[3]{a}}{\frac{1}{c}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   14. Applied times-frac 10.6

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}}\]

   15. Simplified 10.5

       \[\leadsto \color{blue}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\]

   if 1.854315912301685e+89 < b_2

   1. Initial program 58.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 2.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.86870705061354704 \cdot 10^{41}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.4521535344124743 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.8543159123016849 \cdot 10^{89}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))