NMSE problem 3.2.1

Average Error: 34.2 → 6.9

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.19915098118719145 \cdot 10^{121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.4150755057481537 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}\\ \mathbf{elif}\;b_2 \le 3.7711811459025421 \cdot 10^{84}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

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 <= -1.1991509811871915e+121)) {
		temp = (-0.5 * (c / b_2));
	} else {
		double temp_1;
		if ((b_2 <= -3.4150755057481537e-290)) {
			temp_1 = ((1.0 / (sqrt((sqrt(((b_2 * b_2) - (a * c))) - b_2)) / 1.0)) / (sqrt((sqrt(((b_2 * b_2) - (a * c))) - b_2)) / c));
		} else {
			double temp_2;
			if ((b_2 <= 3.771181145902542e+84)) {
				temp_2 = ((-b_2 - sqrt(((b_2 * b_2) - (a * c)))) * (1.0 / a));
			} else {
				temp_2 = ((0.5 * (c / b_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 < -1.1991509811871915e+121

   1. Initial program 60.9

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

   2. Taylor expanded around -inf 2.3

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

   if -1.1991509811871915e+121 < b_2 < -3.4150755057481537e-290

   1. Initial program 32.9

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

   3. Applied flip-- 32.9

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

   4. Simplified 15.4

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

   5. Simplified 15.4

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

   7. Applied *-un-lft-identity 15.4

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

   8. Applied associate-/r* 15.4

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

   9. Simplified 13.7

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

   11. Applied *-un-lft-identity 13.7

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

   12. Applied add-sqr-sqrt 13.8

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

   13. Applied times-frac 13.8

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

   14. Applied *-un-lft-identity 13.8

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

   15. Applied times-frac 14.5

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

   16. Applied associate-/l* 13.8

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

   17. Simplified 8.1

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

   if -3.4150755057481537e-290 < b_2 < 3.771181145902542e+84

   1. Initial program 10.5

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

   3. Applied div-inv 10.6

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

   if 3.771181145902542e+84 < b_2

   1. Initial program 43.2

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

   2. Taylor expanded around inf 4.1

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.19915098118719145 \cdot 10^{121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.4150755057481537 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}\\ \mathbf{elif}\;b_2 \le 3.7711811459025421 \cdot 10^{84}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))