NMSE problem 3.2.1

Average Error: 33.7 → 8.7

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.8085485579241775 \cdot 10^{39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.7444235711350233 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt{1}}}{\sqrt{1}}}{1} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 7.3698697747524969 \cdot 10^{119}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

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 <= -4.8085485579241775e+39)) {
		temp = (-0.5 * (c / b_2));
	} else {
		double temp_1;
		if ((b_2 <= 2.7444235711350233e-95)) {
			temp_1 = (((((a / sqrt(1.0)) / sqrt(1.0)) / 1.0) * (c / (sqrt(((b_2 * b_2) - (a * c))) - b_2))) / a);
		} else {
			double temp_2;
			if ((b_2 <= 7.369869774752497e+119)) {
				temp_2 = (1.0 / (a / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))));
			} else {
				temp_2 = (-2.0 * (b_2 / a));
			}
			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 < -4.8085485579241775e+39

   1. Initial program 57.2

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

   2. Taylor expanded around -inf 3.8

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

   if -4.8085485579241775e+39 < b_2 < 2.7444235711350233e-95

   1. Initial program 23.1

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

   3. Applied flip-- 25.4

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

   4. Simplified 17.8

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

   5. Simplified 17.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 18.1

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}{a}\]

   8. Applied *-un-lft-identity 18.1

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

   9. Applied times-frac 18.1

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

   10. Simplified 17.1

       \[\leadsto \frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} \cdot \color{blue}{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}}{a}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 17.1

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

   13. Applied *-un-lft-identity 17.1

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

   14. Applied sqrt-prod 17.1

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

   15. Applied times-frac 17.1

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

   16. Applied associate-/r* 17.1

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

   17. Applied *-un-lft-identity 17.1

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

   18. Applied sqrt-prod 17.1

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

   19. Applied associate-/r* 17.1

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

   20. Applied frac-times 16.3

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

   21. Simplified 16.3

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

   22. Simplified 15.9

       \[\leadsto \frac{\frac{\frac{\frac{a}{\sqrt{1}}}{\sqrt{1}}}{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
   23. Using strategy rm

   24. Applied clear-num 15.9

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

   25. Applied associate-/r/ 15.8

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

   if 2.7444235711350233e-95 < b_2 < 7.369869774752497e+119

   1. Initial program 5.3

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

   3. Applied clear-num 5.5

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

   if 7.369869774752497e+119 < b_2

   1. Initial program 51.9

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

   3. Applied flip-- 63.8

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

   4. Simplified 62.8

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

   5. Simplified 62.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 62.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}{a}\]

   8. Applied *-un-lft-identity 62.8

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

   9. Applied times-frac 62.8

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

   10. Simplified 62.7

       \[\leadsto \frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} \cdot \color{blue}{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}}{a}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 62.7

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

   13. Applied *-un-lft-identity 62.7

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

   14. Applied sqrt-prod 62.7

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

   15. Applied times-frac 62.7

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

   16. Applied associate-/r* 62.7

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

   17. Applied *-un-lft-identity 62.7

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

   18. Applied sqrt-prod 62.7

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

   19. Applied associate-/r* 62.7

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

   20. Applied frac-times 62.7

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

   21. Simplified 62.7

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

   22. Simplified 62.6

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

   23. Taylor expanded around 0 2.3

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.8085485579241775 \cdot 10^{39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.7444235711350233 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt{1}}}{\sqrt{1}}}{1} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 7.3698697747524969 \cdot 10^{119}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))