Quotient of products

Average Error: 11.1 → 4.6

Time: 3.7s

Precision: binary64

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;\frac{\frac{1 \cdot a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.2503385139216141 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.32763240249908067 \cdot 10^{-275}:\\ \;\;\;\;\frac{\frac{1 \cdot a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.5237975952785877 \cdot 10^{284}:\\ \;\;\;\;\sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double code(double a1, double a2, double b1, double b2) {
	return ((double) (((double) (a1 * a2)) / ((double) (b1 * b2))));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -inf.0)) {
		VAR = ((double) (((double) (((double) (1.0 * a1)) / b1)) / ((double) (b2 / a2))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -6.250338513921614e-284)) {
			VAR_1 = ((double) (1.0 / ((double) (b1 / ((double) (((double) (a1 * a2)) / b2))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 4.3276324024990807e-275)) {
				VAR_2 = ((double) (((double) (((double) (1.0 * a1)) / b1)) / ((double) (b2 / a2))));
			} else {
				double VAR_3;
				if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 5.523797595278588e+284)) {
					VAR_3 = ((double) (((double) sqrt(((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))))) * ((double) sqrt(((double) (((double) (a1 * a2)) / ((double) (b1 * b2))))))));
				} else {
					VAR_3 = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.1
Target	11.6
Herbie	4.6

\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -6.2503385139216141e-284 < (/ (* a1 a2) (* b1 b2)) < 4.32763240249908067e-275

   1. Initial program 16.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied clear-num 17.2

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]
   4. Using strategy rm

   5. Applied times-frac 4.8

      \[\leadsto \frac{1}{\color{blue}{\frac{b1}{a1} \cdot \frac{b2}{a2}}}\]

   6. Applied associate-/r* 4.3

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{b1}{a1}}}{\frac{b2}{a2}}}\]

   7. Simplified 4.1

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot a1}{b1}}}{\frac{b2}{a2}}\]

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -6.2503385139216141e-284

   1. Initial program 0.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied clear-num 0.8

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}}\]
   4. Using strategy rm

   5. Applied associate-/l* 7.6

      \[\leadsto \frac{1}{\color{blue}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}}\]

   if 4.32763240249908067e-275 < (/ (* a1 a2) (* b1 b2)) < 5.5237975952785877e284

   1. Initial program 0.9

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 1.1

      \[\leadsto \color{blue}{\sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}}}\]

   if 5.5237975952785877e284 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 60.0

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied times-frac 8.1

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -inf.0:\\ \;\;\;\;\frac{\frac{1 \cdot a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -6.2503385139216141 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{\frac{b1}{\frac{a1 \cdot a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.32763240249908067 \cdot 10^{-275}:\\ \;\;\;\;\frac{\frac{1 \cdot a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.5237975952785877 \cdot 10^{284}:\\ \;\;\;\;\sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))