quad2m (problem 3.2.1, negative)

Average Error: 34.3 → 6.7

Time: 9.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.0155757731311035 \cdot 10^{+137}:\\ \;\;\;\;\frac{c}{\left(a \cdot \frac{0.5}{\frac{b_2}{c}} - b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \leq 3.872731709600237 \cdot 10^{-232}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 1.9930311221408239 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \end{array}\]

C

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

↓

double code(double a, double b_2, double c) {
	double VAR;
	if ((b_2 <= -2.0155757731311035e+137)) {
		VAR = (c / ((double) (((double) (((double) (a * (0.5 / (b_2 / c)))) - b_2)) - b_2)));
	} else {
		double VAR_1;
		if ((b_2 <= 3.872731709600237e-232)) {
			VAR_1 = (c / ((double) (((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))) - b_2)));
		} else {
			double VAR_2;
			if ((b_2 <= 1.9930311221408239e+59)) {
				VAR_2 = (((double) (((double) -(b_2)) - ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))))) / a);
			} else {
				VAR_2 = ((double) (((double) (0.5 * (c / b_2))) + ((double) ((b_2 / a) * -2.0))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -2.0155757731311035e137

   1. Initial program 62.3

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

   3. Applied flip-- 62.3

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

   4. Simplified 37.0

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

   5. Simplified 37.0

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

   7. Applied *-un-lft-identity 37.0

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

   8. Applied *-un-lft-identity 37.0

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

   9. Applied times-frac 37.0

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

   10. Simplified 37.0

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

   11. Simplified 36.0

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

   12. Taylor expanded around -inf 7.1

       \[\leadsto 1 \cdot \frac{c}{\color{blue}{\left(0.5 \cdot \frac{a \cdot c}{b_2} - b_2\right)} - b_2}\]

   13. Simplified 1.7

       \[\leadsto 1 \cdot \frac{c}{\color{blue}{\left(a \cdot \frac{0.5}{\frac{b_2}{c}} - b_2\right)} - b_2}\]

   if -2.0155757731311035e137 < b_2 < 3.8727317096002367e-232

   1. Initial program 32.1

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

   3. Applied flip-- 32.2

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

   4. Simplified 16.1

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

   5. Simplified 16.1

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

   7. Applied *-un-lft-identity 16.1

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

   8. Applied *-un-lft-identity 16.1

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

   9. Applied times-frac 16.1

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

   10. Simplified 16.1

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

   11. Simplified 9.1

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

   if 3.8727317096002367e-232 < b_2 < 1.99303112214082389e59

   1. Initial program 8.0

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

   if 1.99303112214082389e59 < b_2

   1. Initial program 40.2

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

   2. Taylor expanded around inf 5.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   3. Simplified 5.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.0155757731311035 \cdot 10^{+137}:\\ \;\;\;\;\frac{c}{\left(a \cdot \frac{0.5}{\frac{b_2}{c}} - b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \leq 3.872731709600237 \cdot 10^{-232}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 1.9930311221408239 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))