Quotient of products

Average Error: 11.1 → 5.1

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -9.2690769035523702 \cdot 10^{292}:\\ \;\;\;\;\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -8.2523225915956471 \cdot 10^{-278}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.28170983261853465 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.7030945419074493 \cdot 10^{228}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}\\ \end{array}\]

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double temp;
	if (((b1 * b2) <= -9.26907690355237e+292)) {
		temp = ((a1 * ((cbrt(a2) * cbrt(a2)) / b1)) * (cbrt(a2) / b2));
	} else {
		double temp_1;
		if (((b1 * b2) <= -8.252322591595647e-278)) {
			temp_1 = (a1 * (a2 / (b1 * b2)));
		} else {
			double temp_2;
			if (((b1 * b2) <= 8.281709832618535e-276)) {
				temp_2 = ((a1 / b1) / (b2 / a2));
			} else {
				double temp_3;
				if (((b1 * b2) <= 2.7030945419074493e+228)) {
					temp_3 = (a1 * (a2 / (b1 * b2)));
				} else {
					temp_3 = ((a1 * ((cbrt(a2) * cbrt(a2)) / b1)) * (cbrt(a2) / b2));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.1
Target	11.5
Herbie	5.1

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -9.26907690355237e+292 or 2.7030945419074493e+228 < (* b1 b2)

   1. Initial program 19.3

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

   3. Applied associate-/l* 18.9

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

   5. Applied div-inv 18.9

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

   6. Simplified 18.8

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

   8. Applied add-cube-cbrt 18.9

      \[\leadsto a1 \cdot \frac{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}{b1 \cdot b2}\]

   9. Applied times-frac 7.5

      \[\leadsto a1 \cdot \color{blue}{\left(\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \frac{\sqrt[3]{a2}}{b2}\right)}\]

   10. Applied associate-*r* 3.4

       \[\leadsto \color{blue}{\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}}\]

   if -9.26907690355237e+292 < (* b1 b2) < -8.252322591595647e-278 or 8.281709832618535e-276 < (* b1 b2) < 2.7030945419074493e+228

   1. Initial program 5.2

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

   3. Applied associate-/l* 5.2

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

   5. Applied div-inv 5.5

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

   6. Simplified 5.2

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

   if -8.252322591595647e-278 < (* b1 b2) < 8.281709832618535e-276

   1. Initial program 47.3

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

   3. Applied associate-/l* 48.3

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

   5. Applied *-un-lft-identity 48.3

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

   6. Applied times-frac 20.7

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

   7. Applied associate-/r* 8.5

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

   8. Simplified 8.5

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -9.2690769035523702 \cdot 10^{292}:\\ \;\;\;\;\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -8.2523225915956471 \cdot 10^{-278}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 8.28170983261853465 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.7030945419074493 \cdot 10^{228}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}\right) \cdot \frac{\sqrt[3]{a2}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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