quad2p (problem 3.2.1, positive)

Average Error: 33.7 → 9.0

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 8.0203295214211515 \cdot 10^{-175}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 42194588.455395833:\\ \;\;\;\;1 \cdot \frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]

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 <= -7.700313305414632e+138)) {
		temp = (1.0 * ((0.5 * (c / b_2)) - (2.0 * (b_2 / a))));
	} else {
		double temp_1;
		if ((b_2 <= 8.020329521421151e-175)) {
			temp_1 = (1.0 * ((-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a));
		} else {
			double temp_2;
			if ((b_2 <= 42194588.45539583)) {
				temp_2 = (1.0 * (((0.0 + (a * c)) / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))) / a));
			} else {
				temp_2 = (1.0 * (-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 < -7.700313305414632e+138

   1. Initial program 57.3

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

   3. Applied *-un-lft-identity 57.3

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

   4. Applied *-un-lft-identity 57.3

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

   5. Applied times-frac 57.3

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

   6. Simplified 57.3

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

   7. Taylor expanded around -inf 2.9

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

   if -7.700313305414632e+138 < b_2 < 8.020329521421151e-175

   1. Initial program 10.4

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

   3. Applied *-un-lft-identity 10.4

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

   4. Applied *-un-lft-identity 10.4

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

   5. Applied times-frac 10.4

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

   6. Simplified 10.4

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

   if 8.020329521421151e-175 < b_2 < 42194588.45539583

   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 *-un-lft-identity 31.5

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

   4. Applied *-un-lft-identity 31.5

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

   5. Applied times-frac 31.5

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

   6. Simplified 31.5

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

   8. Applied flip-+ 31.5

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

   9. Simplified 17.1

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

   if 42194588.45539583 < b_2

   1. Initial program 55.9

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

   3. Applied *-un-lft-identity 55.9

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

   4. Applied *-un-lft-identity 55.9

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

   5. Applied times-frac 55.9

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

   6. Simplified 55.9

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

   7. Taylor expanded around inf 5.9

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 8.0203295214211515 \cdot 10^{-175}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 42194588.455395833:\\ \;\;\;\;1 \cdot \frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))