quad2p (problem 3.2.1, positive)

Average Error: 34.3 → 6.3

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.86885461377155539 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.43974566180272646 \cdot 10^{-260}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 2.4235533085097852 \cdot 10^{67}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

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

↓

double code(double a, double b_2, double c) {
	double temp;
	if ((b_2 <= -2.8688546137715554e+112)) {
		temp = ((0.5 * (c / b_2)) - (2.0 * (b_2 / a)));
	} else {
		double temp_1;
		if ((b_2 <= -1.4397456618027265e-260)) {
			temp_1 = ((-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a);
		} else {
			double temp_2;
			if ((b_2 <= 2.4235533085097852e+67)) {
				temp_2 = (1.0 * (c / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))));
			} else {
				temp_2 = (-0.5 * (c / b_2));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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 < -2.8688546137715554e+112

   1. Initial program 50.4

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

   2. Taylor expanded around -inf 3.9

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

   if -2.8688546137715554e+112 < b_2 < -1.4397456618027265e-260

   1. Initial program 7.9

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

   if -1.4397456618027265e-260 < b_2 < 2.4235533085097852e+67

   1. Initial program 29.0

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

   3. Applied flip-+ 29.0

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

   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 13.8

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

   10. Applied *-un-lft-identity 13.8

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

   11. Applied *-un-lft-identity 13.8

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

   12. Applied times-frac 13.8

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

   13. Applied associate-/l* 13.8

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

   14. Simplified 9.3

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 9.3

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

   17. Applied *-un-lft-identity 9.3

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

   18. Applied times-frac 9.3

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

   19. Applied add-sqr-sqrt 9.3

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

   20. Applied times-frac 9.3

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

   21. Simplified 9.3

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

   22. Simplified 8.9

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

   if 2.4235533085097852e+67 < b_2

   1. Initial program 58.0

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

   2. Taylor expanded around inf 3.0

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

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.86885461377155539 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.43974566180272646 \cdot 10^{-260}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 2.4235533085097852 \cdot 10^{67}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))