quad2p (problem 3.2.1, positive)

Average Error: 34.2 → 10.3

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.16299820169664 \cdot 10^{-118}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 1.37297999284257527 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \left(\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

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 <= -3.16299820169664e-118)) {
		temp = (1.0 * ((0.5 * (c / b_2)) - (2.0 * (b_2 / a))));
	} else {
		double temp_1;
		if ((b_2 <= 1.3729799928425753e+84)) {
			temp_1 = (1.0 * ((1.0 / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))) * c));
		} else {
			temp_1 = (-0.5 * (c / b_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 3 regimes

2. if b_2 < -3.16299820169664e-118

   1. Initial program 25.3

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

   3. Applied flip-+ 53.1

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

   4. Simplified 52.7

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

   6. Applied *-un-lft-identity 52.7

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

   7. Applied *-un-lft-identity 52.7

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

   8. Applied times-frac 52.7

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

   9. Simplified 52.7

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

   10. Simplified 52.5

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

   12. Applied *-un-lft-identity 52.5

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

   13. Applied times-frac 52.5

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

   14. Simplified 52.5

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

   15. Simplified 52.5

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

   16. Taylor expanded around -inf 13.7

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

   if -3.16299820169664e-118 < b_2 < 1.3729799928425753e+84

   1. Initial program 26.4

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

   3. Applied flip-+ 27.7

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

   4. Simplified 16.7

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

   6. Applied *-un-lft-identity 16.7

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

   7. Applied *-un-lft-identity 16.7

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

   8. Applied times-frac 16.7

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

   9. Simplified 16.7

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

   10. Simplified 15.2

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

   12. Applied *-un-lft-identity 15.2

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

   13. Applied times-frac 15.2

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

   14. Simplified 15.2

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

   15. Simplified 12.3

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

   17. Applied div-inv 12.3

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

   18. Applied add-cube-cbrt 12.3

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

   19. Applied times-frac 12.2

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

   20. Simplified 12.2

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

   21. Simplified 12.1

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

   if 1.3729799928425753e+84 < b_2

   1. Initial program 58.6

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

   2. Taylor expanded around inf 2.9

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.16299820169664 \cdot 10^{-118}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 1.37297999284257527 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \left(\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))