quad2p (problem 3.2.1, positive)

Average Error: 34.4 → 6.7

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.95155870856424 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + b_2 \cdot \frac{-2}{a}\\ \mathbf{elif}\;b_2 \leq -2.8405866436241524 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.9930311221408239 \cdot 10^{+59}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + a \cdot \frac{-0.5}{\frac{b_2}{c}}\right)}\\ \end{array}\]

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 <= -1.95155870856424e+129)) {
		VAR = ((double) (((double) (0.5 * (c / b_2))) + ((double) (b_2 * (-2.0 / a)))));
	} else {
		double VAR_1;
		if ((b_2 <= -2.8405866436241524e-198)) {
			VAR_1 = (((double) (((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))) - b_2)) / a);
		} else {
			double VAR_2;
			if ((b_2 <= 1.9930311221408239e+59)) {
				VAR_2 = (((double) -(c)) / ((double) (b_2 + ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))))));
			} else {
				VAR_2 = (((double) -(c)) / ((double) (b_2 + ((double) (b_2 + ((double) (a * (-0.5 / (b_2 / c)))))))));
			}
			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 < -1.95155870856424e129

   1. Initial program 56.1

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

   2. Simplified 56.1

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

   3. Taylor expanded around -inf 2.6

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

   4. Simplified 2.8

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

   if -1.95155870856424e129 < b_2 < -2.8405866436241524e-198

   1. Initial program 7.2

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

   2. Simplified 7.2

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

   if -2.8405866436241524e-198 < b_2 < 1.99303112214082389e59

   1. Initial program 27.1

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

   2. Simplified 27.1

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

   4. Applied flip-- 27.2

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

   5. Simplified 16.8

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

   6. Simplified 16.8

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

   8. Applied distribute-rgt-neg-out 16.8

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

   9. Applied distribute-frac-neg 16.8

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

   10. Applied distribute-frac-neg 16.8

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

   11. Simplified 10.3

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

   if 1.99303112214082389e59 < b_2

   1. Initial program 57.9

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

   2. Simplified 57.9

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

   4. Applied flip-- 57.9

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

   5. Simplified 30.4

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

   6. Simplified 30.4

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

   8. Applied distribute-rgt-neg-out 30.4

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

   9. Applied distribute-frac-neg 30.4

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

   10. Applied distribute-frac-neg 30.4

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

   11. Simplified 27.1

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

   12. Taylor expanded around inf 8.0

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

   13. Simplified 3.8

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.95155870856424 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + b_2 \cdot \frac{-2}{a}\\ \mathbf{elif}\;b_2 \leq -2.8405866436241524 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.9930311221408239 \cdot 10^{+59}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \left(b_2 + a \cdot \frac{-0.5}{\frac{b_2}{c}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))